Prism Graph

Author

Eric W. Weisstein
October 21, 2024

This notebook downloaded from http://mathworld.wolfram.com/notebooks/GraphTheory/PrismGraph.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/PrismGraph.html.

©2024 Wolfram Research, Inc. except for portions noted otherwise

Definition in terms of (Subscript[K, 2])□(Subscript[C, 2])

ToEntity/@(SimpleGraph[EdgeList[#]]&/@Table[GraphProduct[CompleteGraph[2],CycleGraph[n],"Cartesian"],{n,2,7}])

Special cases

With[{cl="Prism"},TextGrid[Sort[{cl/.GraphData[#,"NotationRules"],GraphData[#,"Name"]}&/@DeleteCases[GraphData[cl],{cl,_}]],Dividers->All]]

Circulant

ToEntity/@Table[CirculantGraph[2n,{2,n}],{n,3,10}]

Embedding types

Graphs.m

GraphicsGrid[Partition[Table[PrismGraph[n,VertexSize->{"Scaled",.05},System`VertexStyle->Red,System`EdgeStyle->Black],{n,3,12}],5],Dividers->All]

Graphs.m "SquareEmbedding"

GraphicsGrid[Partition[Table[PrismGraph[n,GraphLayout->"SquareEmbedding",VertexSize->{"Scaled",.05},System`VertexStyle->Red,System`EdgeStyle->Black],{n,3,12}],5],Dividers->All]

GraphData

All

GraphicsGrid[Partition[Table[GraphData[{"Prism",n}],{n,3,25}],5],Dividers->All]//StyleGraphs

(Subscript[Y, 3])

GraphicsGrid[Partition[GraphData["Y3","Graph","All"]//StyleGraphs,UpTo[4]]]

GraphData["Y3","EmbeddingClasses"]//Column

(Subscript[Y, 4])

GraphicsGrid[Partition[GraphData["Y4","Graph","All"]//StyleGraphs,UpTo[5]]]

GraphData["Y4","EmbeddingClasses"]//Column

(Subscript[Y, 5])

GraphicsGrid[Partition[GraphData["Y5","Graph","All"]//StyleGraphs,UpTo[5]]]

GraphData["Y5","EmbeddingClasses"]//Column

UnitDistance

GraphicsGrid[Partition[Table[Take[GraphData[{"Prism",n},"Graph","UnitDistance"],UpTo[1]],{n,3,25}],5],Dividers->All]//StyleGraphs

Embeddings

<<MathWorld`Graphs`

{"Prism", 3}

Names

GraphData[{"Prism",3},"StandardName"]

GraphData[{"Prism",3},"StandardNames"]

Primary

GraphData[{"Prism",3}]//StyleGraphs[#,ImageSize->Small]&

Read and Wilson

All

GraphData[{"Prism",3},"Graph","All"]//StyleGraphs

Inexact

Select[GraphData[{"Prism",3},"Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData[{"Prism",3},"EdgeList"]]

Default embedding

GraphPlot[GraphData[{"Prism",3},"Edges"],GraphLayout->"RadialEmbedding"]

GraphPlot@GraphData[{"Prism",3},"Edges"]

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

Degenerate

GraphData[{"Prism",3},"Graph","Degenerate"]//StyleGraphs

Bilateral

GraphData[{"Prism",3},"Graph","Bilateral"]//StyleGraphs

Grid[DeleteCases[Monitor[{g=#,BilateralGraphEmbeddingTypes[#],GraphSymmetryGroup[#]}&/@GraphData[{"Prism",3},"Graph","All"],g],{_,_,TransformationGroup[{}]}],Dividers->All,Alignment->{{Center,{Left}}}]

Circular

GraphData[{"Prism",3},"Graph","Circular"]//StyleGraphs

Planar

GraphData[{"Prism",3},"Planar"]

GraphData[{"Prism",3},"Graph","Planar"]//StyleGraphs

Tutte embeddings

faces=ReverseSort@SymmetricallyDistinctFaces[GraphData[{"Prism",3}]]

Tally[ReverseSort[Length/@faces]]

IGLayoutTutte[GraphData[{"Prism",3}],"OuterFace"->#]&/@faces

gs=Annotate[#,{VertexLabels->None,ImageSize->Tiny}]&/@{}

RecognizeGraph/@gs

Tutte embedding construction

GraphPlot[RotateGraph[]]

v=Join[];
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

MinimalCrossing

GraphData[{"Prism",3},{"RCN","CN"}]

GraphData[{"Prism",3},"Graph","MinimalCrossing"]//StyleGraphs

Circulant

GraphData[{"Prism",3},"Circulant"]

GraphData[{"Prism",3},"Graph","Circulant"]//StyleGraphs

Circulant construction

CirculantGraph1@@@CirculantIndices[GraphData[{"Prism",3}]]

gs=CirculantGraph1@@@Cases[GraphData[{"Prism",3},"StandardNames"],{"Circulant",i_}:>i]

Halin

GraphData[{"Prism",3},"Halin"]

GraphData[{"Prism",3},"Graph","Halin"]//StyleGraphs

Rigid

GraphData[{"Prism",3},"Rigid"]

RigidGraphQ[GraphData[{"Prism",3}]]

GraphData[{"Prism",3},"Graph","Rigid"]//StyleGraphs

Flexible

GraphData[{"Prism",3},"Flexible"]

GraphData[{"Prism",3},"Graph","Flexible"]//StyleGraphs

InfinitesimallyRigidFrameworkQ[#,GraphEmbedding[#]]&/@%

LCF

GraphData[{"Prism",3},"LCF"]

GraphData[{"Prism",3},"HamiltonianCycleCount"]

GraphData[{"Prism",3},"LCFSignature"]

GraphData[{"Prism",3},"Graph","LCF"]//StyleGraphs

Tally[LCFNotation[#][[2]]&/@%]

LCF compute

Length[FindHamiltonianCycle[GraphData[{"Prism",3}],All][[All,All,1]]]//Timing

Tally[(lcf=LCF[GraphData[{"Prism",3}],All])[[All,2]]]//Timing

LCFGraph@@@lcf//StyleGraphs

gs=Flatten[{
(* all LCF's with order >1, sorted by order and with bilateral graphs first *)
Join[DeleteCases[#,{_Missing,_}][[All,1]],Cases[#,{_Missing,_}][[All,2]]]&/@Map[{BilateralLCFEmbedding[#],#}&,Map[LCFGraph@@#&,SplitBy[DeleteCases[lcf,{_,1}],Last],{2}],{2}],
(* order-1 bilateral only *)
DeleteMissing[BilateralLCFEmbedding/@LCFGraph@@@Cases[lcf,{_,1}]]}]

Length[gs]

UnitDistance

GraphData[{"Prism",3},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"Prism",3}],Debug->True]//Timing

GraphData[{"Prism",3},"Graph","UnitDistance"]//StyleGraphs

Select[%,!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&]

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData[{"Prism",3},"VertexList"],GraphData[{"Prism",3},"EdgeList"]]]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]]]&/@Complement@@(GraphData[{"Prism",3},"Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData[{"Prism",3}],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData[{"Prism",3}],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance by FindSubgraphUnitDistanceEmbedding

{{"Prism",3},l=FindSubgraphUnitDistanceEmbedding[GraphData[{"Prism",3}],TimeConstraint->300,"Results"->All]//Timing}

gs=SortBy[UniqueEmbeddings[l],Replace[RectilinearCrossingCount[#],Infinity->0]&]

Matchstick

GraphData[{"Prism",3},"Matchstick"]

GraphData[{"Prism",3},"Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData[{"Prism",3},"Edges"]]

3D All

GraphData[{"Prism",3},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"Prism",3},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"Prism",3},"Polyhedral"]

GraphData[{"Prism",3},"Graph",{"3D","Polyhedron"}]

Graphics3D[{FaceForm[Green,Red],GraphData[{"Prism",3},"Polyhedron"]}]

3D Canonical

GraphData[{"Prism",3},"Graph",{"3D","Canonical"}]

Graphics3D[{Opacity[.5],Red,GraphData[{"Prism",3},"Polyhedron","Canonical"],Yellow,Sphere[]}]

3D Canonical construction

Graphics3D[{FaceForm[Green,Red],poly=GraphPolyhedron[GraphData[{"Prism",3}],1]}]

(can=FindCanonicalPolyhedron[poly,WorkingPrecision->50,MaxIterations->50000,PrecisionGoal->20,"ProgressInterval"->1000,"TangentFactor"->1/10,"PlanarFactor"->1/10,Debug->True])//Timing

SimplePolyhedronQ[can]

Graphics3D[{Opacity[.5],Red,can,Yellow,Sphere[]}]

(lens=Mean/@Split[Sort[PolyhedronEdgeLengths[can]],Abs[#1-#2]<1.*^-10&])//N

v=PolyhedronCoordinates[can];

colors=With[{n=Length[lens]},Table[Hue[i/n],{i,0,n-1}]]

Graphics3D[{Thick,Sequence@@Nearest[Thread[lens->colors],EuclideanDistance@@v[[#]]],Line[v[[#]]]}&/@PolyhedronEdges[can],Boxed->False]

g=PolyhedronGraph[can]

{"Prism", 4} == "CubicalGraph"

GraphData[{"Prism",4},"StandardName"]

{"Prism", 5}

Names

GraphData[{"Prism",5},"StandardName"]

GraphData[{"Prism",5},"StandardNames"]

Primary

GraphData[{"Prism",5}]//StyleGraphs[#,ImageSize->Small]&

Read and Wilson

Default embedding

GraphPlot@GraphData[{"Prism",5},"Edges"]

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

All

GraphData[{"Prism",5},"Graph","All"]//StyleGraphs

Inexact

Select[GraphData[{"Prism",5},"Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData[{"Prism",5},"EdgeList"]]

Degenerate

GraphData[{"Prism",5},"Graph","Degenerate"]//StyleGraphs

IGraph

GraphData[{"Prism",5},"Graph","IGraph"]//StyleGraphs

Bilateral

GraphData[{"Prism",5},"Graph","Bilateral"]//StyleGraphs

Grid[DeleteCases[Monitor[{g=#,BilateralGraphEmbeddingTypes[#],GraphSymmetryGroup[#]}&/@GraphData[{"Prism",5},"Graph","All"],g],{_,_,TransformationGroup[{}]}],Dividers->All,Alignment->{{Center,{Left}}}]

Circular

GraphData[{"Prism",5},"Graph","Circular"]//StyleGraphs

Circular construction

GraphData[{"Prism",5},"Hamiltonian"]

GraphData[{"Prism",5},"HamiltonianCycleCount"]

Length[hc=FindHamiltonianCycle[GraphData[{"Prism",5}],All][[All,All,1]]]//Timing

(circ=CircularEmbeddingFromHamiltonianCycle[GraphData[{"Prism",5}],#]&/@hc);//Timing

sym={#,GraphSymmetryGroup[#]}&/@circ;//Timing

Tally[sym[[All,-1]]]

Cases[sym,{gn_,TransformationGroup[XXX]}:>gn]

Length[bilat=DeleteDuplicatesBy[
DeleteMissing[BilaterallyRotatedGraph/@circ],
CircularCanonicalGraphEmbedding]]//Timing

gs=bilat

Length[quad=DeleteDuplicatesBy[Union@Select[bilat,BilateralGraphEmbeddingQ[#,Axis->"X"]&],CircularCanonicalGraphEmbedding[#,4]&]]

quad

Planar

GraphData[{"Prism",5},"Planar"]

GraphData[{"Prism",5},"Graph","Planar"]//StyleGraphs

Tutte embeddings

faces=ReverseSort@SymmetricallyDistinctFaces[GraphData[{"Prism",5}]]

Tally[ReverseSort[Length/@faces]]

IGLayoutTutte[GraphData[{"Prism",5}],"OuterFace"->#]&/@faces

gs=Annotate[#,{VertexLabels->None,ImageSize->Tiny}]&/@{}

Tutte embedding construction

GraphPlot[RotateGraph[,Pi]]

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

MinimalCrossing

GraphData[{"Prism",5},{"RCN","CN"}]

GraphData[{"Prism",5},"Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

g=RotateGraph[MinimalCrossingGraph[GraphData[{"Prism",5}],"CrossingNumber"->-1]]

RecognizeGraph[g]

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData[{"Prism",5},"NotationRules","Circulant"]

Cases[GraphData[{"Prism",5},"StandardNames"],{"Circulant",_}]

GraphData[{"Prism",5},"Graph","Circulant"]//StyleGraphs

Circulant construction

gs=CirculantGraph1@@@CirculantIndices[GraphData[{"Prism",5}],All]

Halin

GraphData[{"Prism",5},"Halin"]

GraphData[{"Prism",5},"Graph","Halin"]//StyleGraphs

LCF

GraphData[{"Prism",5},"LCF"]

GraphData[{"Prism",5},"LCFSignature"]

GraphData[{"Prism",5},"Graph","LCF"]//StyleGraphs

LCF compute

Length[FindHamiltonianCycle[GraphData[{"Prism",5}],All][[All,All,1]]]//Timing

Tally[(lcf=LCF[GraphData[{"Prism",5}],All])[[All,2]]]//Timing

LCFGraph@@@lcf//StyleGraphs

gs=Flatten[{
(* all LCF's with order >1, sorted by order and with bilateral graphs first *)
Join[DeleteCases[#,{_Missing,_}][[All,1]],Cases[#,{_Missing,_}][[All,2]]]&/@Map[{BilateralLCFEmbedding[#],#}&,Map[LCFGraph@@#&,SplitBy[DeleteCases[lcf,{_,1}],Last],{2}],{2}],
(* order-1 bilateral only *)
DeleteMissing[BilateralLCFEmbedding/@LCFGraph@@@Cases[lcf,{_,1}]]}]

UnitDistance

GraphData[{"Prism",5},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"Prism",5}],Debug->True]//Timing

GraphData[{"Prism",5},"Graph","UnitDistance"]//StyleGraphs

Select[%,!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&]

Select[UnitDistanceEmbeddingRefine/@Complement@@(GraphData[{"Prism",5},"Graph",#]&/@{"All","UnitDistance"}),UnitDistanceGraphEmbeddingQ]

FindUnitDistanceEmbedding[GraphData[{"Prism",5}],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

UnitDistance FindSubgraphUnitDistanceEmbedding

{{"Prism",5},l=FindSubgraphUnitDistanceEmbedding[GraphData[{"Prism",5}],TimeConstraint->300,"Results"->All]//Timing}

gs=UniqueEmbeddings[l]

gs={,,RotateGraph[,{6,1}],RotateGraph[,{10,5}]}

Matchstick

GraphData[{"Prism",5},"Matchstick"]

GraphData[{"Prism",5},"Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData[{"Prism",5},"Edges"]]

3D All

GraphData[{"Prism",5},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"Prism",5},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"Prism",5},"Polyhedral"]

GraphData[{"Prism",5},"Graph",{"3D","Polyhedron"}]

Graphics3D[{FaceForm[Green,Red],GraphData[{"Prism",5},"Polyhedron"]}]

3D Canonical

GraphData[{"Prism",5},"Graph",{"3D","Canonical"}]

Graphics3D[{Opacity[.5],Red,GraphData[{"Prism",5},"Polyhedron","Canonical"],Yellow,Sphere[]}]

{"Prism", 6}

Names

GraphData[{"Prism",6},"StandardName"]

GraphData[{"Prism",6},"StandardNames"]

Primary

GraphData[{"Prism",6}]//StyleGraphs[#,ImageSize->Small]&

Read and Wilson

Default embedding

GraphPlot@GraphData[{"Prism",6},"Edges"]

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

All

GraphData[{"Prism",6},"Graph","All"]//StyleGraphs

Inexact

Select[GraphData[{"Prism",6},"Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData[{"Prism",6},"EdgeList"]]

Degenerate

GraphData[{"Prism",6},"Graph","Degenerate"]//StyleGraphs

Bilateral

GraphData[{"Prism",6},"Graph","Bilateral"]//StyleGraphs

Planar

GraphData[{"Prism",6},"Planar"]

GraphData[{"Prism",6},"Graph","Planar"]//StyleGraphs

Tutte embeddings

faces=ReverseSort@SymmetricallyDistinctFaces[GraphData[{"Prism",6}]]

Tally[ReverseSort[Length/@faces]]

IGLayoutTutte[GraphData[{"Prism",6}],"OuterFace"->#]&/@faces

gs=Annotate[#,{VertexLabels->None,ImageSize->Tiny}]&/@{}

Tutte embedding construction 4

GraphPlot[RotateGraph[,Pi]]

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

MinimalCrossing

GraphData[{"Prism",6},{"RCN","CN"}]

GraphData[{"Prism",6},"Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

g=RotateGraph[MinimalCrossingGraph[GraphData[{"Prism",6}],"CrossingNumber"->-1]]

RecognizeGraph[g]

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData[{"Prism",6},"Circulant"]

GraphData[{"Prism",6},"Graph","Circulant"]//StyleGraphs

Halin

GraphData[{"Prism",6},"Halin"]

GraphData[{"Prism",6},"Graph","Halin"]//StyleGraphs

LCF

GraphData[{"Prism",6},"LCF"]

GraphData[{"Prism",6},"LCFSignature"]

GraphData[{"Prism",6},"Graph","LCF"]//StyleGraphs

LCF compute

Length[FindHamiltonianCycle[GraphData[{"Prism",6}],All][[All,All,1]]]//Timing

Tally[(lcf=LCF[GraphData[{"Prism",6}],All])[[All,2]]]//Timing

LCFGraph@@@lcf//StyleGraphs

gs=Flatten[{
(* all LCF's with order >1, sorted by order and with bilateral graphs first *)
Join[DeleteCases[#,{_Missing,_}][[All,1]],Cases[#,{_Missing,_}][[All,2]]]&/@Map[{BilateralLCFEmbedding[#],#}&,Map[LCFGraph@@#&,SplitBy[DeleteCases[lcf,{_,1}],Last],{2}],{2}],
(* order-1 bilateral only *)
DeleteMissing[BilateralLCFEmbedding/@LCFGraph@@@Cases[lcf,{_,1}]]}]

UnitDistance

GraphData[{"Prism",6},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"Prism",6}],Debug->True]//Timing

GraphData[{"Prism",6},"Graph","UnitDistance"]//StyleGraphs

Select[%,!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&]

Select[UnitDistanceEmbeddingRefine/@Complement@@(GraphData[{"Prism",6},"Graph",#]&/@{"All","UnitDistance"}),UnitDistanceGraphEmbeddingQ]

FindUnitDistanceEmbedding[GraphData[{"Prism",6}],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

UnitDistance FindSubgraphUnitDistanceEmbedding

{{"Prism",6},l=FindSubgraphUnitDistanceEmbedding[GraphData[{"Prism",6}],TimeConstraint->300,"Results"->All]//Timing}

gs=UniqueEmbeddings[l]

Matchstick

GraphData[{"Prism",6},"Matchstick"]

GraphData[{"Prism",6},"Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData[{"Prism",6},"Edges"]]

3D All

GraphData[{"Prism",6},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"Prism",6},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"Prism",6},"Polyhedral"]

GraphData[{"Prism",6},"Graph",{"3D","Polyhedron"}]

Graphics3D[{FaceForm[Green,Red],GraphData[{"Prism",6},"Polyhedron"]}]

3D Canonical

GraphData[{"Prism",6},"Graph",{"3D","Canonical"}]

Graphics3D[{Opacity[.5],Red,GraphData[{"Prism",6},"Polyhedron","Canonical"],Yellow,Sphere[]}]

{"Prism", 7}

Names

GraphData[{"Prism",7},"StandardName"]

GraphData[{"Prism",7},"StandardNames"]

Primary

GraphData[{"Prism",7}]//StyleGraphs[#,ImageSize->Small]&

Read and Wilson

All

GraphData[{"Prism",7},"Graph","All"]//StyleGraphs

Inexact

Select[GraphData[{"Prism",7},"Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData[{"Prism",7},"EdgeList"]]

Default embedding

GraphPlot@GraphData[{"Prism",7},"Edges"]

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

Degenerate

GraphData[{"Prism",7},"Graph","Degenerate"]//StyleGraphs

Bilateral

GraphData[{"Prism",7},"Graph","Bilateral"]//StyleGraphs

Circular

GraphData[{"Prism",7},"Graph","Circular"]//StyleGraphs

Planar

GraphData[{"Prism",7},"Planar"]

GraphData[{"Prism",7},"Graph","Planar"]//StyleGraphs

Tutte embeddings

faces=ReverseSort@SymmetricallyDistinctFaces[GraphData[{"Prism",7}]]

Tally[ReverseSort[Length/@faces]]

IGLayoutTutte[GraphData[{"Prism",7}],"OuterFace"->#]&/@faces

gs=Annotate[#,{VertexLabels->None,ImageSize->Tiny}]&/@{}

RecognizeGraph/@gs

Tutte embedding construction

GraphPlot[RotateGraph[]]

v=Join[];
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

MinimalCrossing

GraphData[{"Prism",7},{"RCN","CN"}]

GraphData[{"Prism",7},"Graph","MinimalCrossing"]//StyleGraphs

Circulant

GraphData[{"Prism",7},"Circulant"]

GraphData[{"Prism",7},"Graph","Circulant"]//StyleGraphs

Halin

GraphData[{"Prism",7},"Halin"]

GraphData[{"Prism",7},"Graph","Halin"]//StyleGraphs

Rigid

GraphData[{"Prism",7},"Rigid"]

GraphData[{"Prism",7},"Graph","Rigid"]//StyleGraphs

Flexible

GraphData[{"Prism",7},"Flexible"]

GraphData[{"Prism",7},"Graph","Flexible"]//StyleGraphs

LCF

GraphData[{"Prism",7},"LCF"]

GraphData[{"Prism",7},"HamiltonianCycleCount"]

GraphData[{"Prism",7},"LCFSignature"]

GraphData[{"Prism",7},"Graph","LCF"]//StyleGraphs

Tally[LCFNotation[#][[2]]&/@%]

LCF compute

Length[FindHamiltonianCycle[GraphData[{"Prism",7}],All][[All,All,1]]]//Timing

Tally[(lcf=LCF[GraphData[{"Prism",7}],All])[[All,2]]]//Timing

lcf

LCFGraph@@@lcf//StyleGraphs

gs=Flatten[{
(* all LCF's with order >1, sorted by order and with bilateral graphs first *)
Join[DeleteCases[#,{_Missing,_}][[All,1]],Cases[#,{_Missing,_}][[All,2]]]&/@Map[{BilateralLCFGraph[#],#}&,Map[LCFGraph@@#&,SplitBy[DeleteCases[lcf,{_,1}],Last],{2}],{2}],
(* order-1 bilateral only *)
DeleteMissing[BilateralLCFGraph/@LCFGraph@@@Cases[lcf,{_,1}]]}]

Length[gs]

UnitDistance

GraphData[{"Prism",7},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"Prism",7}],Debug->True]//Timing

GraphData[{"Prism",7},"Graph","UnitDistance"]//StyleGraphs

Select[%,!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&]

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData[{"Prism",7},"VertexList"],GraphData[{"Prism",7},"EdgeList"]]]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]]]&/@Complement@@(GraphData[{"Prism",7},"Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance construction

g0=RotateGraph[,Pi/2]

List@@@EdgeList[g0]

GraphPlot@

v={{0,h+1},{-x,y+1},{-3/2,1},{-1/2,1},{1/2,1},{3/2,1},{x,y+1},{0,h},{-x,y},{-3/2,0},{-1/2,0},{1/2,0},{3/2,0},{x,y}};
eqn=UnitEdgeEquations[v,g0]//FullSimplify//Union

Solve[eqn,{x,y,h},PositiveReals]//FullSimplify

Solve[Join[eqn,{h==1/4}],{x,y,h},PositiveReals][[1]]

Maximize[{h,{(x^2)+(((h-y))^2)==1,(((-(3/2)+x))^2)+(y^2)==1}},{x,y,h}]//FullSimplify

N[((Sqrt[7])/2)]

Manipulate[Module[{soln=NSolve[Join[{(x^2)+(((h-y))^2)==1,(((-(3/2)+x))^2)+(y^2)==1}],{x,y},Reals],
v={{0,h+1},{-x,y+1},{-3/2,1},{-1/2,1},{1/2,1},{3/2,1},{x,y+1},{0,h},{-x,y},{-3/2,0},{-1/2,0},{1/2,0},{3/2,0},{x,y}},edges={{1,2},{1,7},{1,8},{2,3},{2,9},{3,4},{3,10},{4,5},{4,11},{5,6},{5,12},{6,7},{6,13},{7,14},{8,9},{8,14},{9,10},{10,11},{11,12},{12,13},{13,14}}},
Graph[Range[14],UndirectedEdge@@@edges,VertexCoordinates->(v/.soln[[1]])]
],{h,-Sqrt[3]/2+.01,((Sqrt[7])/2)-.01}]

gsoln=Function[f,Graph[Range@Length@v,EdgeList[g0],VertexCoordinates->(v/.#)]&/@Solve[Append[eqn,h==f],{x,y,h},Reals]]/@{-1/2,0,1,Sqrt[3]/2,((Sqrt[7])/2)}

Select[Flatten[gsoln],UnitDistanceGraphEmbeddingQ]

gs={,,}

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData[{"Prism",7}],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData[{"Prism",7}],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance by FindSubgraphUnitDistanceEmbedding

{{"Prism",7},l=FindSubgraphUnitDistanceEmbedding[GraphData[{"Prism",7}],TimeConstraint->300,"Results"->All]//Timing}

gs=SortBy[UniqueEmbeddings[l],Replace[RectilinearCrossingCount[#],Infinity->0]&]

Matchstick

GraphData[{"Prism",7},"Matchstick"]

GraphData[{"Prism",7},"Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData[{"Prism",7},"Edges"]]

3D All

GraphData[{"Prism",7},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"Prism",7},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"Prism",7},"Polyhedral"]

GraphData[{"Prism",7},"Graph",{"3D","Polyhedron"}]

Graphics3D[{FaceForm[Green,Red],GraphData[{"Prism",7},"Polyhedron"]}]

3D Canonical

GraphData[{"Prism",7},"Graph",{"3D","Canonical"}]

Graphics3D[{Opacity[.5],Red,GraphData[{"Prism",7},"Polyhedron","Canonical"],Yellow,Sphere[]}]

3D Canonical construction

Graphics3D[{FaceForm[Green,Red],poly=GraphPolyhedron[GraphData[{"Prism",7}],1]}]

(can=FindCanonicalPolyhedron[poly,WorkingPrecision->50,MaxIterations->50000,PrecisionGoal->20,"ProgressInterval"->1000,"TangentFactor"->1/10,"PlanarFactor"->1/10,Debug->True])//Timing

SimplePolyhedronQ[can]

If[%===False,Quit[]]

Graphics3D[{Opacity[.5],Red,can,Yellow,Sphere[]}]

(lens=Mean/@Split[Sort[PolyhedronEdgeLengths[can]],Abs[#1-#2]<1.*^-10&])//N

v=PolyhedronCoordinates[can];

colors=With[{n=Length[lens]},Table[Hue[i/n],{i,0,n-1}]]

Graphics3D[{Thick,Sequence@@Nearest[Thread[lens->colors],EuclideanDistance@@v[[#]]],Line[v[[#]]]}&/@PolyhedronEdges[can],Boxed->False]

g=PolyhedronGraph[can]

3D Canonical construction from PolyhedralEmbedding

g=GraphData[{"Prism",7},"Graph","Planar"][[1]]

Show[g2=PolyhedralEmbedding[g,"Circular"->True,"Scale"->3],Boxed->True]//Quiet

With[
{faces=PlanarFaceList[Graph[EdgeList[g2]]],v=GraphEmbedding@g2},
Graphics3D[{Point[{0,0,0}],Opacity[.5],Sphere[],FaceForm[Green,Red],
poly=Polyhedron[1.2v,faces]}]]

(can=FindCanonicalPolyhedron[poly,WorkingPrecision->100,MaxIterations->20000,PrecisionGoal->20,"ProgressInterval"->100,"TangentFactor"->1/5,"PlanarFactor"->1/5,Debug->True])//Timing

SimplePolyhedronQ[can]

If[%===False,Quit[]]

Graphics3D[{Opacity[.5],Red,can,Yellow,Sphere[]}]

(lens=Mean/@Split[Sort[PolyhedronEdgeLengths[can]],Abs[#1-#2]<1.*^-10&])//N

v=PolyhedronCoordinates[can];

colors=With[{n=Length[lens]},Table[Hue[i/n],{i,0,n-1}]]

Graphics3D[{Thick,Sequence@@Nearest[Thread[lens->colors],EuclideanDistance@@v[[#]]],Line[v[[#]]]}&/@PolyhedronEdges[can],Boxed->False]

g=PolyhedronGraph[can]

{"Prism", 8}

Names

GraphData[{"Prism",8},"StandardName"]

GraphData[{"Prism",8},"StandardNames"]

Primary

GraphData[{"Prism",8}]//StyleGraphs[#,ImageSize->Small]&

Read and Wilson

All

GraphData[{"Prism",8},"Graph","All"]//StyleGraphs

Inexact

Select[GraphData[{"Prism",8},"Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData[{"Prism",8},"EdgeList"]]

Default embedding

GraphPlot@GraphData[{"Prism",8},"Edges"]

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

Degenerate

GraphData[{"Prism",8},"Graph","Degenerate"]//StyleGraphs

HoneycombToroidal construction

GraphData[{"Prism",8},"NotationRules","HoneycombToroidal"]

HoneycombToroidalGraph[1,16,3]

RecognizeGraph[%]

GeneralizedHoneycombGraph construction

With[{n=16},Cases[Table[{k,g=GeneralizedHoneycombGraph[n,k],RecognizeGraph@g},{k,n-1}],{__,{"Prism",8}}]]//Timing

g=GeneralizedHoneycombGraph[16,5]

RecognizeGraph@g

PrismGraph[8,GraphLayout->"HoneycombToroidalEmbedding"]

RecognizeGraph@%

Bilateral

GraphData[{"Prism",8},"Graph","Bilateral"]//StyleGraphs

Circular

GraphData[{"Prism",8},"Graph","Circular"]//StyleGraphs

Planar

GraphData[{"Prism",8},"Planar"]

GraphData[{"Prism",8},"Graph","Planar"]//StyleGraphs

Tutte embeddings

faces=ReverseSort@SymmetricallyDistinctFaces[GraphData[{"Prism",8}]]

Tally[ReverseSort[Length/@faces]]

IGLayoutTutte[GraphData[{"Prism",8}],"OuterFace"->#]&/@faces

gs=Annotate[#,{VertexLabels->None,ImageSize->Tiny}]&/@{}

RecognizeGraph/@gs

Tutte embedding construction

GraphPlot[RotateGraph[]]

v=Join[];
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

MinimalCrossing

GraphData[{"Prism",8},{"RCN","CN"}]

GraphData[{"Prism",8},"Graph","MinimalCrossing"]//StyleGraphs

Circulant

GraphData[{"Prism",8},"Circulant"]

GraphData[{"Prism",8},"Graph","Circulant"]//StyleGraphs

Halin

GraphData[{"Prism",8},"Halin"]

GraphData[{"Prism",8},"Graph","Halin"]//StyleGraphs

Rigid

GraphData[{"Prism",8},"Rigid"]

GraphData[{"Prism",8},"Graph","Rigid"]//StyleGraphs

Flexible

GraphData[{"Prism",8},"Flexible"]

GraphData[{"Prism",8},"Graph","Flexible"]//StyleGraphs

LCF

GraphData[{"Prism",8},"LCF"]

GraphData[{"Prism",8},"HamiltonianCycleCount"]

GraphData[{"Prism",8},"LCFSignature"]

GraphData[{"Prism",8},"Graph","LCF"]//StyleGraphs

Tally[LCFNotation[#][[2]]&/@%]

LCF compute

Length[FindHamiltonianCycle[GraphData[{"Prism",8}],All][[All,All,1]]]//Timing

Tally[(lcf=LCF[GraphData[{"Prism",8}],All])[[All,2]]]//Timing

clf

LCFGraph@@@lcf//StyleGraphs

gs=Flatten[{
(* all LCF's with order >1, sorted by order and with bilateral graphs first *)
Join[DeleteCases[#,{_Missing,_}][[All,1]],Cases[#,{_Missing,_}][[All,2]]]&/@Map[{BilateralLCFGraph[#],#}&,Map[LCFGraph@@#&,SplitBy[DeleteCases[lcf,{_,1}],Last],{2}],{2}],
(* order-1 bilateral only *)
DeleteMissing[BilateralLCFGraph/@LCFGraph@@@Cases[lcf,{_,1}]]}]

Length[gs]

UnitDistance

GraphData[{"Prism",8},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"Prism",8}],Debug->True]//Timing

GraphData[{"Prism",8},"Graph","UnitDistance"]//StyleGraphs

Select[%,!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&]

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData[{"Prism",8},"VertexList"],GraphData[{"Prism",8},"EdgeList"]]]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]]]&/@Complement@@(GraphData[{"Prism",8},"Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

LabelGraph@

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData[{"Prism",8}],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData[{"Prism",8}],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance by FindSubgraphUnitDistanceEmbedding

{{"Prism",8},l=FindSubgraphUnitDistanceEmbedding[GraphData[{"Prism",8}],TimeConstraint->300,"Results"->All]//Timing}

gs=SortBy[UniqueEmbeddings[l],Replace[RectilinearCrossingCount[#],Infinity->0]&]

gs={,,,,,,,,};

Matchstick

GraphData[{"Prism",8},"Matchstick"]

GraphData[{"Prism",8},"Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData[{"Prism",8},"Edges"]]

3D All

GraphData[{"Prism",8},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"Prism",8},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"Prism",8},"Polyhedral"]

GraphData[{"Prism",8},"Graph",{"3D","Polyhedron"}]

Graphics3D[{FaceForm[Green,Red],GraphData[{"Prism",8},"Polyhedron"]}]

3D Canonical

GraphData[{"Prism",8},"Graph",{"3D","Canonical"}]

Graphics3D[{Opacity[.5],Red,GraphData[{"Prism",8},"Polyhedron","Canonical"],Yellow,Sphere[]}]

3D Canonical construction

Graphics3D[{FaceForm[Green,Red],poly=GraphPolyhedron[GraphData[{"Prism",8}],1]}]

(can=FindCanonicalPolyhedron[poly,WorkingPrecision->50,MaxIterations->50000,PrecisionGoal->20,"ProgressInterval"->1000,"TangentFactor"->1/10,"PlanarFactor"->1/10,Debug->True])//Timing

SimplePolyhedronQ[can]

Graphics3D[{Opacity[.5],Red,can,Yellow,Sphere[]}]

(lens=Mean/@Split[Sort[PolyhedronEdgeLengths[can]],Abs[#1-#2]<1.*^-10&])//N

v=PolyhedronCoordinates[can];

colors=With[{n=Length[lens]},Table[Hue[i/n],{i,0,n-1}]]

Graphics3D[{Thick,Sequence@@Nearest[Thread[lens->colors],EuclideanDistance@@v[[#]]],Line[v[[#]]]}&/@PolyhedronEdges[can],Boxed->False]

g=PolyhedronGraph[can]

{"Prism", 9}

Names

GraphData[{"Prism",9},"StandardName"]

GraphData[{"Prism",9},"StandardNames"]

Primary

GraphData[{"Prism",9}]//StyleGraphs[#,ImageSize->Small]&

Read and Wilson

All

GraphData[{"Prism",9},"Graph","All"]//StyleGraphs

Inexact

Select[GraphData[{"Prism",9},"Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData[{"Prism",9},"EdgeList"]]

Default embedding

GraphPlot@GraphData[{"Prism",9},"Edges"]

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

Degenerate

GraphData[{"Prism",9},"Graph","Degenerate"]//StyleGraphs

Bilateral

GraphData[{"Prism",9},"Graph","Bilateral"]//StyleGraphs

Circular

GraphData[{"Prism",9},"Graph","Circular"]//StyleGraphs

Planar

GraphData[{"Prism",9},"Planar"]

GraphData[{"Prism",9},"Graph","Planar"]//StyleGraphs

Tutte embeddings

faces=ReverseSort@SymmetricallyDistinctFaces[GraphData[{"Prism",9}]]

Tally[ReverseSort[Length/@faces]]

IGLayoutTutte[GraphData[{"Prism",9}],"OuterFace"->#]&/@faces

gs=Annotate[#,{VertexLabels->None,ImageSize->Tiny}]&/@{}

RecognizeGraph/@gs

Tutte embedding construction

GraphPlot[RotateGraph[]]

v=Join[];
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

MinimalCrossing

GraphData[{"Prism",9},{"RCN","CN"}]

GraphData[{"Prism",9},"Graph","MinimalCrossing"]//StyleGraphs

Circulant

GraphData[{"Prism",9},"Circulant"]

GraphData[{"Prism",9},"Graph","Circulant"]//StyleGraphs

Halin

GraphData[{"Prism",9},"Halin"]

GraphData[{"Prism",9},"Graph","Halin"]//StyleGraphs

Rigid

GraphData[{"Prism",9},"Rigid"]

GraphData[{"Prism",9},"Graph","Rigid"]//StyleGraphs

Flexible

GraphData[{"Prism",9},"Flexible"]

GraphData[{"Prism",9},"Graph","Flexible"]//StyleGraphs

LCF

GraphData[{"Prism",9},"LCF"]

GraphData[{"Prism",9},"HamiltonianCycleCount"]

GraphData[{"Prism",9},"LCFSignature"]

GraphData[{"Prism",9},"Graph","LCF"]//StyleGraphs

Tally[LCFNotation[#][[2]]&/@%]

LCF compute

Length[FindHamiltonianCycle[GraphData[{"Prism",9}],All][[All,All,1]]]//Timing

Tally[(lcf=LCF[GraphData[{"Prism",9}],All])[[All,2]]]//Timing

lcf

LCFGraph@@@lcf//StyleGraphs

gs=Flatten[{
(* all LCF's with order >1, sorted by order and with bilateral graphs first *)
Join[DeleteCases[#,{_Missing,_}][[All,1]],Cases[#,{_Missing,_}][[All,2]]]&/@Map[{BilateralLCFGraph[#],#}&,Map[LCFGraph@@#&,SplitBy[DeleteCases[lcf,{_,1}],Last],{2}],{2}],
(* order-1 bilateral only *)
DeleteMissing[BilateralLCFGraph/@LCFGraph@@@Cases[lcf,{_,1}]]}]

Length[gs]

UnitDistance

GraphData[{"Prism",9},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"Prism",9}],Debug->True]//Timing

GraphData[{"Prism",9},"Graph","UnitDistance"]//StyleGraphs

Select[%,!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&]

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData[{"Prism",9},"VertexList"],GraphData[{"Prism",9},"EdgeList"]]]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]]]&/@Complement@@(GraphData[{"Prism",9},"Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData[{"Prism",9}],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData[{"Prism",9}],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance by FindSubgraphUnitDistanceEmbedding

{{"Prism",9},l=FindSubgraphUnitDistanceEmbedding[GraphData[{"Prism",9}],TimeConstraint->300,"Results"->All]//Timing}

gs=SortBy[UniqueEmbeddings[l],Replace[RectilinearCrossingCount[#],Infinity->0]&]

gs={,,};

Matchstick

GraphData[{"Prism",9},"Matchstick"]

GraphData[{"Prism",9},"Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData[{"Prism",9},"Edges"]]

3D All

GraphData[{"Prism",9},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"Prism",9},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"Prism",9},"Polyhedral"]

GraphData[{"Prism",9},"Graph",{"3D","Polyhedron"}]

Graphics3D[{FaceForm[Green,Red],GraphData[{"Prism",9},"Polyhedron"]}]

3D Canonical

GraphData[{"Prism",9},"Graph",{"3D","Canonical"}]

Graphics3D[{Opacity[.5],Red,GraphData[{"Prism",9},"Polyhedron","Canonical"],Yellow,Sphere[]}]

3D Canonical construction

Graphics3D[{FaceForm[Green,Red],poly=GraphPolyhedron[GraphData[{"Prism",9}],-1]}]

(can=FindCanonicalPolyhedron[poly,WorkingPrecision->50,MaxIterations->50000,PrecisionGoal->20,"ProgressInterval"->1000,"TangentFactor"->1/10,"PlanarFactor"->1/10,Debug->True])//Timing

SimplePolyhedronQ[can]

Graphics3D[{Opacity[.5],Red,can,Yellow,Sphere[]}]

(lens=Mean/@Split[Sort[PolyhedronEdgeLengths[can]],Abs[#1-#2]<1.*^-10&])//N

v=PolyhedronCoordinates[can];

colors=With[{n=Length[lens]},Table[Hue[i/n],{i,0,n-1}]]

Graphics3D[{Thick,Sequence@@Nearest[Thread[lens->colors],EuclideanDistance@@v[[#]]],Line[v[[#]]]}&/@PolyhedronEdges[can],Boxed->False]

g=PolyhedronGraph[can]

{"Prism", 10}

Names

GraphData[{"Prism",10},"StandardName"]

GraphData[{"Prism",10},"StandardNames"]

Primary

GraphData[{"Prism",10}]//StyleGraphs[#,ImageSize->Small]&

Read and Wilson construction

All

GraphData[{"Prism",10},"Graph","All"]//StyleGraphs

Inexact

Select[GraphData[{"Prism",10},"Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData[{"Prism",10},"EdgeList"]]

Default embedding

GraphPlot@GraphData[{"Prism",10},"Edges"]

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

Degenerate

GraphData[{"Prism",10},"Graph","Degenerate"]//StyleGraphs

GeneralizedHoneycombGraph construction

With[{n=20},Cases[Table[{k,g=GeneralizedHoneycombGraph[n,k],RecognizeGraph@g},{k,n-1}],{__,{"Prism",_}}]]//Timing

g=GeneralizedHoneycombGraph[20,7]

RecognizeGraph@g

PrismGraph[10,GraphLayout->"HoneycombToroidalEmbedding"]

RecognizeGraph@%

Bilateral

GraphData[{"Prism",10},"Graph","Bilateral"]//StyleGraphs

Circular

GraphData[{"Prism",10},"Graph","Circular"]//StyleGraphs

Planar

GraphData[{"Prism",10},"Planar"]

GraphData[{"Prism",10},"Graph","Planar"]//StyleGraphs

Tutte embeddings

faces=ReverseSort@SymmetricallyDistinctFaces[GraphData[{"Prism",10}]]

Tally[ReverseSort[Length/@faces]]

IGLayoutTutte[GraphData[{"Prism",10}],"OuterFace"->#]&/@faces

gs=Annotate[#,{VertexLabels->None,ImageSize->Tiny}]&/@{}

RecognizeGraph/@gs

Tutte embedding construction

GraphPlot[RotateGraph[]]

v=Join[];
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

MinimalCrossing

GraphData[{"Prism",10},{"RCN","CN"}]

GraphData[{"Prism",10},"Graph","MinimalCrossing"]//StyleGraphs

Circulant

GraphData[{"Prism",10},"Circulant"]

GraphData[{"Prism",10},"Graph","Circulant"]//StyleGraphs

Halin

GraphData[{"Prism",10},"Halin"]

GraphData[{"Prism",10},"Graph","Halin"]//StyleGraphs

Rigid

GraphData[{"Prism",10},"Rigid"]

GraphData[{"Prism",10},"Graph","Rigid"]//StyleGraphs

Flexible

GraphData[{"Prism",10},"Flexible"]

GraphData[{"Prism",10},"Graph","Flexible"]//StyleGraphs

LCF

GraphData[{"Prism",10},"LCF"]

GraphData[{"Prism",10},"HamiltonianCycleCount"]

GraphData[{"Prism",10},"LCFSignature"]

GraphData[{"Prism",10},"Graph","LCF"]//StyleGraphs

Tally[LCFNotation[#][[2]]&/@%]

LCF compute

Length[FindHamiltonianCycle[GraphData[{"Prism",10}],All][[All,All,1]]]//Timing

Tally[(lcf=LCF[GraphData[{"Prism",10}],All])[[All,2]]]//Timing

lcf

LCFGraph@@@lcf//StyleGraphs

gs=Flatten[{
(* all LCF's with order >1, sorted by order and with bilateral graphs first *)
Join[DeleteCases[#,{_Missing,_}][[All,1]],Cases[#,{_Missing,_}][[All,2]]]&/@Map[{BilateralLCFGraph[#],#}&,Map[LCFGraph@@#&,SplitBy[DeleteCases[lcf,{_,1}],Last],{2}],{2}],
(* order-1 bilateral only *)
DeleteMissing[BilateralLCFGraph/@LCFGraph@@@Cases[lcf,{_,1}]]}]

Length[gs]

UnitDistance

GraphData[{"Prism",10},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"Prism",10}],Debug->True]//Timing

GraphData[{"Prism",10},"Graph","UnitDistance"]//StyleGraphs

Select[%,!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&]

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData[{"Prism",10},"VertexList"],GraphData[{"Prism",10},"EdgeList"]]]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]]]&/@Complement@@(GraphData[{"Prism",10},"Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData[{"Prism",10}],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData[{"Prism",10}],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance by FindSubgraphUnitDistanceEmbedding

{{"Prism",10},l=FindSubgraphUnitDistanceEmbedding[GraphData[{"Prism",10}],TimeConstraint->300,"Results"->All]//Timing}

gs=SortBy[UniqueEmbeddings[l],Replace[RectilinearCrossingCount[#],Infinity->0]&]

gs={,,,,,,,,,,,,,};

Matchstick

GraphData[{"Prism",10},"Matchstick"]

GraphData[{"Prism",10},"Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData[{"Prism",10},"Edges"]]

3D All

GraphData[{"Prism",10},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"Prism",10},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"Prism",10},"Polyhedral"]

GraphData[{"Prism",10},"Graph",{"3D","Polyhedron"}]

Graphics3D[{FaceForm[Green,Red],GraphData[{"Prism",10},"Polyhedron"]}]

3D Canonical

GraphData[{"Prism",10},"Graph",{"3D","Canonical"}]

Graphics3D[{Opacity[.5],Red,GraphData[{"Prism",10},"Polyhedron","Canonical"],Yellow,Sphere[]}]

3D Canonical construction

Graphics3D[{FaceForm[Green,Red],poly=GraphPolyhedron[GraphData[{"Prism",10}],1]}]

(can=FindCanonicalPolyhedron[poly,WorkingPrecision->50,MaxIterations->50000,PrecisionGoal->20,"ProgressInterval"->1000,"TangentFactor"->1/10,"PlanarFactor"->1/10,Debug->True])//Timing

SimplePolyhedronQ[can]

Graphics3D[{Opacity[.5],Red,can,Yellow,Sphere[]}]

(lens=Mean/@Split[Sort[PolyhedronEdgeLengths[can]],Abs[#1-#2]<1.*^-10&])//N

v=PolyhedronCoordinates[can];

colors=With[{n=Length[lens]},Table[Hue[i/n],{i,0,n-1}]]

Graphics3D[{Thick,Sequence@@Nearest[Thread[lens->colors],EuclideanDistance@@v[[#]]],Line[v[[#]]]}&/@PolyhedronEdges[can],Boxed->False]

g=PolyhedronGraph[can]

{"Prism", 11}

Names

GraphData[{"Prism",11},"StandardName"]

GraphData[{"Prism",11},"StandardNames"]

Primary

GraphData[{"Prism",11}]//StyleGraphs[#,ImageSize->Small]&

Read and Wilson construction

All

GraphData[{"Prism",11},"Graph","All"]//StyleGraphs

Inexact

Select[GraphData[{"Prism",11},"Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData[{"Prism",11},"EdgeList"]]

Default embedding

GraphPlot@GraphData[{"Prism",11},"Edges"]

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

Degenerate

GraphData[{"Prism",11},"Graph","Degenerate"]//StyleGraphs

DegenerateGraphEmbeddingTypes/@GraphData[{"Prism",11},"Graph","Degenerate"]

Perspective

GraphData[{"Prism",11},"Graph","Perspective"]//StyleGraphs

Perspective construction

PolyhedronGraph[GraphData[{"Prism",11},"Polyhedron"]]

Manipulate[PolyhedronProjectionGraph[GraphData[{"Prism",11},"Polyhedron"],{x,y,z}],{x,-1,1},{y,-1,1},{z,-1,1}]

PolyhedronProjectionGraph[GraphData[{"Prism",11},"Polyhedron"],{0,.01,.01}]

PolyhedronProjectionGraph[GraphData[{"Prism",11},"Polyhedron"],{0,0,1}]

PolyhedronProjectionGraph[GraphData[{"Prism",11},"Polyhedron"],{0,1,0}]

PolyhedronProjectionGraph[GraphData[{"Prism",11},"Polyhedron"],{1,0.01,0}]

PolyhedronProjectionGraph[GraphData[{"Prism",11},"Polyhedron"],{1,1,0}]

PolyhedronProjectionGraph[GraphData[{"Prism",11},"Polyhedron"],{1,0,1,-Pi/2}]

PolyhedronProjectionGraph[GraphData[{"Prism",11},"Polyhedron"],{1,1,1}]

Bilateral

GraphData[{"Prism",11},"Graph","Bilateral"]//StyleGraphs

Circular

GraphData[{"Prism",11},"Graph","Circular"]//StyleGraphs

Planar

GraphData[{"Prism",11},"Planar"]

GraphData[{"Prism",11},"Graph","Planar"]//StyleGraphs

Tutte embeddings

faces=ReverseSort@SymmetricallyDistinctFaces[GraphData[{"Prism",11}]]

Tally[ReverseSort[Length/@faces]]

IGLayoutTutte[GraphData[{"Prism",11}],"OuterFace"->#]&/@faces

gs=Annotate[#,{VertexLabels->None,ImageSize->Tiny}]&/@{}

RecognizeGraph/@gs

MinimalCrossing

GraphData[{"Prism",11},{"RCN","CN"}]

GraphData[{"Prism",11},"Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

(cn=CrossingNumber[GraphData[{"Prism",11}],TimeConstraint->1])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"Prism",11}],TimeConstraint->10])//AbsoluteTiming

(g=MinimalCrossingGraph[GraphData[{"Prism",11}],"CrossingNumber"->cn])//PolyhedralGraphQ

faces=ReverseSort@SymmetricallyDistinctFaces[g]

Tally[ReverseSort[Length/@faces]]

IGLayoutTutte[g,"OuterFace"->#]&/@faces

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData[{"Prism",11}],"CrossingNumber"->cn,RandomSeed->n]],{n,50}],Max[OuterFace[#]]<=GraphData[{"Prism",11},"V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData[{"Prism",11}],"CrossingNumber"->cn,RandomSeed->i],{i,50}]//Union

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData[{"Prism",11},"Circulant"]

GraphData[{"Prism",11},"Graph","Circulant"]//StyleGraphs

Complement graph

GraphData[{"Prism",11},"ComplementGraph","Name"]

Line graph

GraphData[{"Prism",11},"Line"]

LineGraphQ[GraphData[{"Prism",11}]]//Timing

RootGraph

Select[GraphData[All],GraphData[#,"LineGraph","Name"]=== {"Prism",11}&]

RootGraph[GraphData[{"Prism",11}]]

Halin

GraphData[{"Prism",11},"Halin"]

GraphData[{"Prism",11},"Graph","Halin"]//StyleGraphs

Rigid

GraphData[{"Prism",11},"Rigid"]

GraphData[{"Prism",11},"Graph","Rigid"]//StyleGraphs

Flexible

GraphData[{"Prism",11},"Flexible"]

GraphData[{"Prism",11},"Graph","Flexible"]//StyleGraphs

LCF

GraphData[{"Prism",11},"LCF"]

GraphData[{"Prism",11},"HamiltonianCycleCount"]

GraphData[{"Prism",11},"LCFSignature"]

GraphData[{"Prism",11},"LCFNotations"]

GraphData[{"Prism",11},"Graph","LCF"]//StyleGraphs

Tally[LCFNotation[#][[2]]&/@%]

LCF compute all

Divisors[GraphData[{"Prism",11},"V"]]

Length[FindHamiltonianCycle[GraphData[{"Prism",11}],All][[All,All,1]]]//Timing

Tally[(lcf=LCF[GraphData[{"Prism",11}],All])[[All,2]]]//Timing

LCFGraph@@@DeleteCases[lcf,{_,1}]//StyleGraphs

Tally[LCFNotation[#][[-1]]&/@(gs=NiceLCFGraphs[lcf,2,1])]//Timing

Length[gs]

gs

UnitDistance

GraphData[{"Prism",11},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"Prism",11}],Debug->True]//Timing

GraphData[{"Prism",11},"Graph","UnitDistance"]//StyleGraphs

Select[%,!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&]

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData[{"Prism",11},"VertexList"],GraphData[{"Prism",11},"EdgeList"]],MaxIterations->500]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]],MaxIterations->500]&/@Complement@@(GraphData[{"Prism",11},"Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData[{"Prism",11}],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData[{"Prism",11}],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance by FindSubgraphUnitDistanceEmbedding

{{"Prism",11},l=FindSubgraphUnitDistanceEmbedding[GraphData[{"Prism",11}],TimeConstraint->300,"Results"->All]//Timing}

gs=SortBy[UniqueEmbeddings[l],Replace[RectilinearCrossingCount[#],Infinity->0]&]

gs={,};

Matchstick

GraphData[{"Prism",11},"Matchstick"]

GraphData[{"Prism",11},"Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData[{"Prism",11},"Edges"]]

3D All

GraphData[{"Prism",11},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"Prism",11},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"Prism",11},"Polyhedral"]

GraphData[{"Prism",11},"Graph",{"3D","Polyhedron"}]

Graphics3D[{FaceForm[Green,Red],GraphData[{"Prism",11},"Polyhedron"]}]

3D Canonical

GraphData[{"Prism",11},"Graph",{"3D","Canonical"}]

Graphics3D[{Opacity[.5],Red,GraphData[{"Prism",11},"Polyhedron","Canonical"],Yellow,Sphere[]}]

{"Prism", 12}

Names

GraphData[{"Prism",12},"StandardName"]

GraphData[{"Prism",12},"StandardNames"]

Primary

GraphData[{"Prism",12}]//StyleGraphs[#,ImageSize->Small]&

Read and Wilson construction

All

GraphData[{"Prism",12},"Graph","All"]//StyleGraphs

Inexact

Select[GraphData[{"Prism",12},"Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData[{"Prism",12},"EdgeList"]]

Default embedding

GraphPlot@GraphData[{"Prism",12},"Edges"]

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

Degenerate

GraphData[{"Prism",12},"Graph","Degenerate"]//StyleGraphs

Bilateral

GraphData[{"Prism",12},"Graph","Bilateral"]//StyleGraphs

Circular

GraphData[{"Prism",12},"Graph","Circular"]//StyleGraphs

Planar

GraphData[{"Prism",12},"Planar"]

GraphData[{"Prism",12},"Graph","Planar"]//StyleGraphs

Tutte embeddings

faces=ReverseSort@SymmetricallyDistinctFaces[GraphData[{"Prism",12}]]

Tally[ReverseSort[Length/@faces]]

IGLayoutTutte[GraphData[{"Prism",12}],"OuterFace"->#]&/@faces

gs=Annotate[#,{VertexLabels->None,ImageSize->Tiny}]&/@{}

RecognizeGraph/@gs

MinimalCrossing

GraphData[{"Prism",12},{"RCN","CN"}]

GraphData[{"Prism",12},"Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

(cn=CrossingNumber[GraphData[{"Prism",12}],TimeConstraint->10])//AbsoluteTiming

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData[{"Prism",12}],"CrossingNumber"->cn,RandomSeed->n]],{n,50}],Max[OuterFace[#]]<=GraphData[{"Prism",12},"V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData[{"Prism",12}],"CrossingNumber"->cn,RandomSeed->i],{i,50}]//Union

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData[{"Prism",12},"Circulant"]

GraphData[{"Prism",12},"Graph","Circulant"]//StyleGraphs

Halin

GraphData[{"Prism",12},"Halin"]

GraphData[{"Prism",12},"Graph","Halin"]//StyleGraphs

Rigid

GraphData[{"Prism",12},"Rigid"]

GraphData[{"Prism",12},"Graph","Rigid"]//StyleGraphs

Flexible

GraphData[{"Prism",12},"Flexible"]

GraphData[{"Prism",12},"Graph","Flexible"]//StyleGraphs

LCF

GraphData[{"Prism",12},"LCF"]

GraphData[{"Prism",12},"HamiltonianCycleCount"]

GraphData[{"Prism",12},"LCFSignature"]

GraphData[{"Prism",12},"Graph","LCF"]//StyleGraphs

Tally[LCFNotation[#][[2]]&/@%]

LCF compute

Length[FindHamiltonianCycle[GraphData[{"Prism",12}],All][[All,All,1]]]//Timing

Tally[(lcf=LCF[GraphData[{"Prism",12}],All])[[All,2]]]//Timing

LCFGraph@@@lcf//StyleGraphs

Tally[LCFNotation[#][[-1]]&/@(gs=NiceLCFGraphs[lcf,2,1])]//Timing

Length[gs]

gs

UnitDistance

GraphData[{"Prism",12},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"Prism",12}],Debug->True]//Timing

GraphData[{"Prism",12},"Graph","UnitDistance"]//StyleGraphs

Select[%,!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&]

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData[{"Prism",12},"VertexList"],GraphData[{"Prism",12},"EdgeList"]]]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]]]&/@Complement@@(GraphData[{"Prism",12},"Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

ContractCoincidentVertices@

RecognizeGraph@%

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData[{"Prism",12}],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData[{"Prism",12}],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance by FindSubgraphUnitDistanceEmbedding

{{"Prism",12},l=FindSubgraphUnitDistanceEmbedding[GraphData[{"Prism",12}],TimeConstraint->300,"Results"->All]//Timing}

gs=SortBy[UniqueEmbeddings[l],Replace[RectilinearCrossingCount[#],Infinity->0]&]

gs={,,,,,,,,,,,,,,,,,,,,,,};

Matchstick

GraphData[{"Prism",12},"Matchstick"]

GraphData[{"Prism",12},"Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData[{"Prism",12},"Edges"]]

3D All

GraphData[{"Prism",12},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"Prism",12},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"Prism",12},"Polyhedral"]

GraphData[{"Prism",12},"Graph",{"3D","Polyhedron"}]

Graphics3D[{FaceForm[Green,Red],GraphData[{"Prism",12},"Polyhedron"]}]

3D Canonical

GraphData[{"Prism",12},"Graph",{"3D","Canonical"}]

Graphics3D[{Opacity[.5],Red,GraphData[{"Prism",12},"Polyhedron","Canonical"],Yellow,Sphere[]}]

{"Prism", 13}

Names

GraphData[{"Prism",13},"StandardName"]

GraphData[{"Prism",13},"StandardNames"]

Primary

GraphData[{"Prism",13}]//StyleGraphs[#,ImageSize->Small]&

Read and Wilson construction

All

GraphData[{"Prism",13},"Graph","All"]//StyleGraphs

Inexact

Select[GraphData[{"Prism",13},"Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData[{"Prism",13},"EdgeList"]]

Default embedding

GraphPlot@GraphData[{"Prism",13},"Edges"]

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

Degenerate

GraphData[{"Prism",13},"Graph","Degenerate"]//StyleGraphs

Bilateral

GraphData[{"Prism",13},"Graph","Bilateral"]//StyleGraphs

Circular

GraphData[{"Prism",13},"Graph","Circular"]//StyleGraphs

Planar

GraphData[{"Prism",13},"Planar"]

GraphData[{"Prism",13},"Graph","Planar"]//StyleGraphs

Tutte embeddings

faces=ReverseSort@SymmetricallyDistinctFaces[GraphData[{"Prism",13}]]

Tally[ReverseSort[Length/@faces]]

IGLayoutTutte[GraphData[{"Prism",13}],"OuterFace"->#]&/@faces

gs=Annotate[#,{VertexLabels->None,ImageSize->Tiny}]&/@{}

RecognizeGraph/@gs

Tutte embedding construction

GraphPlot[RotateGraph[]]

v=Join[];
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

MinimalCrossing

GraphData[{"Prism",13},{"RCN","CN"}]

GraphData[{"Prism",13},"Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

(cn=CrossingNumber[GraphData[{"Prism",13}],TimeConstraint->10])//AbsoluteTiming

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData[{"Prism",13}],"CrossingNumber"->cn,RandomSeed->n]],{n,50}],Max[OuterFace[#]]<=GraphData[{"Prism",13},"V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData[{"Prism",13}],"CrossingNumber"->cn,RandomSeed->i],{i,50}]//Union

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData[{"Prism",13},"Circulant"]

GraphData[{"Prism",13},"Graph","Circulant"]//StyleGraphs

Halin

GraphData[{"Prism",13},"Halin"]

GraphData[{"Prism",13},"Graph","Halin"]//StyleGraphs

Rigid

GraphData[{"Prism",13},"Rigid"]

GraphData[{"Prism",13},"Graph","Rigid"]//StyleGraphs

Flexible

GraphData[{"Prism",13},"Flexible"]

GraphData[{"Prism",13},"Graph","Flexible"]//StyleGraphs

LCF

GraphData[{"Prism",13},"LCF"]

GraphData[{"Prism",13},"HamiltonianCycleCount"]

GraphData[{"Prism",13},"LCFSignature"]

GraphData[{"Prism",13},"Graph","LCF"]//StyleGraphs

Tally[LCFNotation[#][[2]]&/@%]

LCF compute

Length[FindHamiltonianCycle[GraphData[{"Prism",13}],All][[All,All,1]]]//Timing

Tally[(lcf=LCF[GraphData[{"Prism",13}],All])[[All,2]]]//Timing

LCFGraph@@@lcf//StyleGraphs

Tally[LCFNotation[#][[-1]]&/@(gs=NiceLCFGraphs[lcf,2,1])]//Timing

Length[gs]

gs

UnitDistance

GraphData[{"Prism",13},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"Prism",13}],Debug->True]//Timing

GraphData[{"Prism",13},"Graph","UnitDistance"]//StyleGraphs

Select[%,!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&]

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData[{"Prism",13},"VertexList"],GraphData[{"Prism",13},"EdgeList"]]]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]]]&/@Complement@@(GraphData[{"Prism",13},"Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData[{"Prism",13}],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData[{"Prism",13}],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance by FindSubgraphUnitDistanceEmbedding

{{"Prism",13},l=FindSubgraphUnitDistanceEmbedding[GraphData[{"Prism",13}],TimeConstraint->300,"Results"->All]//Timing}

gs=SortBy[UniqueEmbeddings[l],Replace[RectilinearCrossingCount[#],Infinity->0]&]

gs={,,,,};

Matchstick

GraphData[{"Prism",13},"Matchstick"]

GraphData[{"Prism",13},"Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData[{"Prism",13},"Edges"]]

3D All

GraphData[{"Prism",13},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"Prism",13},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"Prism",13},"Polyhedral"]

GraphData[{"Prism",13},"Graph",{"3D","Polyhedron"}]

Graphics3D[{FaceForm[Green,Red],GraphData[{"Prism",13},"Polyhedron"]}]

3D Canonical

GraphData[{"Prism",13},"Graph",{"3D","Canonical"}]

Graphics3D[{Opacity[.5],Red,GraphData[{"Prism",13},"Polyhedron","Canonical"],Yellow,Sphere[]}]

{"Prism", 14}

Names

GraphData[{"Prism",14},"StandardName"]

GraphData[{"Prism",14},"StandardNames"]

Primary

GraphData[{"Prism",14}]//StyleGraphs[#,ImageSize->Small]&

Read and Wilson

All

GraphData[{"Prism",14},"Graph","All"]//StyleGraphs

Inexact

Select[GraphData[{"Prism",14},"Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData[{"Prism",14},"EdgeList"]]

Default embedding

GraphPlot@GraphData[{"Prism",14},"Edges"]

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

Degenerate

GraphData[{"Prism",14},"Graph","Degenerate"]//StyleGraphs

GeneralizedHoneycombGraph construction

With[{n=28},Cases[Table[{k,g=GeneralizedHoneycombGraph[n,k],RecognizeGraph@g},{k,n-1}],{__,{"Prism",_}}]]//Timing

g=GeneralizedHoneycombGraph[28,9]

RecognizeGraph@g

Bilateral

GraphData[{"Prism",14},"Graph","Bilateral"]//StyleGraphs

Circular

GraphData[{"Prism",14},"Graph","Circular"]//StyleGraphs

Planar

GraphData[{"Prism",14},"Planar"]

GraphData[{"Prism",14},"Graph","Planar"]//StyleGraphs

Tutte embeddings

faces=ReverseSort@SymmetricallyDistinctFaces[GraphData[{"Prism",14}]]

Tally[ReverseSort[Length/@faces]]

IGLayoutTutte[GraphData[{"Prism",14}],"OuterFace"->#]&/@faces

gs=Annotate[#,{VertexLabels->None,ImageSize->Tiny}]&/@{}

RecognizeGraph/@gs

Tutte embedding construction

GraphPlot[RotateGraph[]]

v=Join[];
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

MinimalCrossing

GraphData[{"Prism",14},{"RCN","CN"}]

GraphData[{"Prism",14},"Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

(cn=CrossingNumber[GraphData[{"Prism",14}],TimeConstraint->10])//AbsoluteTiming

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData[{"Prism",14}],"CrossingNumber"->cn,RandomSeed->n]],{n,50}],Max[OuterFace[#]]<=GraphData[{"Prism",14},"V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData[{"Prism",14}],"CrossingNumber"->cn,RandomSeed->i],{i,50}]//Union

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData[{"Prism",14},"Circulant"]

GraphData[{"Prism",14},"Graph","Circulant"]//StyleGraphs

Halin

GraphData[{"Prism",14},"Halin"]

GraphData[{"Prism",14},"Graph","Halin"]//StyleGraphs

Rigid

GraphData[{"Prism",14},"Rigid"]

GraphData[{"Prism",14},"Graph","Rigid"]//StyleGraphs

Flexible

GraphData[{"Prism",14},"Flexible"]

GraphData[{"Prism",14},"Graph","Flexible"]//StyleGraphs

LCF

GraphData[{"Prism",14},"LCF"]

GraphData[{"Prism",14},"HamiltonianCycleCount"]

GraphData[{"Prism",14},"LCFSignature"]

GraphData[{"Prism",14},"Graph","LCF"]//StyleGraphs

Tally[LCFNotation[#][[2]]&/@%]

LCF compute

Length[FindHamiltonianCycle[GraphData[{"Prism",14}],All][[All,All,1]]]//Timing

Tally[(lcf=LCF[GraphData[{"Prism",14}],All])[[All,2]]]//Timing

LCFGraph@@@lcf//StyleGraphs

Tally[LCFNotation[#][[-1]]&/@(gs=NiceLCFGraphs[lcf,2,1])]//Timing

Length[gs]

gs

UnitDistance

GraphData[{"Prism",14},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"Prism",14}],Debug->True]//Timing

GraphData[{"Prism",14},"Graph","UnitDistance"]//StyleGraphs

Select[%,!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&]

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData[{"Prism",14},"VertexList"],GraphData[{"Prism",14},"EdgeList"]]]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]]]&/@Complement@@(GraphData[{"Prism",14},"Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData[{"Prism",14}],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData[{"Prism",14}],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance by FindSubgraphUnitDistanceEmbedding

{{"Prism",14},l=FindSubgraphUnitDistanceEmbedding[GraphData[{"Prism",14}],TimeConstraint->300,"Results"->All]//Timing}

gs=SortBy[UniqueEmbeddings[l],Replace[RectilinearCrossingCount[#],Infinity->0]&]

gs={,,,,,,,,,,,,,,,,,,,,,,,,,,};

Matchstick

GraphData[{"Prism",14},"Matchstick"]

GraphData[{"Prism",14},"Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData[{"Prism",14},"Edges"]]

3D All

GraphData[{"Prism",14},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"Prism",14},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"Prism",14},"Polyhedral"]

GraphData[{"Prism",14},"Graph",{"3D","Polyhedron"}]

Graphics3D[{FaceForm[Green,Red],GraphData[{"Prism",14},"Polyhedron"]}]

3D Canonical

GraphData[{"Prism",14},"Graph",{"3D","Canonical"}]

Graphics3D[{Opacity[.5],Red,GraphData[{"Prism",14},"Polyhedron","Canonical"],Yellow,Sphere[]}]

{"Prism", 15}

Names

GraphData[{"Prism",15},"StandardName"]

GraphData[{"Prism",15},"StandardNames"]

Primary

GraphData[{"Prism",15}]//StyleGraphs[#,ImageSize->Small]&

Read and Wilson

All

GraphData[{"Prism",15},"Graph","All"]//StyleGraphs

Inexact

Select[GraphData[{"Prism",15},"Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData[{"Prism",15},"EdgeList"]]

Default embedding

GraphPlot@GraphData[{"Prism",15},"Edges"]

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

Degenerate

GraphData[{"Prism",15},"Graph","Degenerate"]//StyleGraphs

Bilateral

GraphData[{"Prism",15},"Graph","Bilateral"]//StyleGraphs

Circular

GraphData[{"Prism",15},"Graph","Circular"]//StyleGraphs

Planar

GraphData[{"Prism",15},"Planar"]

GraphData[{"Prism",15},"Graph","Planar"]//StyleGraphs

Tutte embeddings

faces=ReverseSort@SymmetricallyDistinctFaces[GraphData[{"Prism",15}]]

Tally[ReverseSort[Length/@faces]]

IGLayoutTutte[GraphData[{"Prism",15}],"OuterFace"->#]&/@faces

gs=Annotate[#,{VertexLabels->None,ImageSize->Tiny}]&/@{}

RecognizeGraph/@gs

MinimalCrossing

GraphData[{"Prism",15},{"RCN","CN"}]

GraphData[{"Prism",15},"Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

(cn=CrossingNumber[GraphData[{"Prism",15}],TimeConstraint->10])//AbsoluteTiming

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData[{"Prism",15}],"CrossingNumber"->cn,RandomSeed->n]],{n,50}],Max[OuterFace[#]]<=GraphData[{"Prism",15},"V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData[{"Prism",15}],"CrossingNumber"->cn,RandomSeed->i],{i,50}]//Union

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData[{"Prism",15},"Circulant"]

GraphData[{"Prism",15},"Graph","Circulant"]//StyleGraphs

Halin

GraphData[{"Prism",15},"Halin"]

GraphData[{"Prism",15},"Graph","Halin"]//StyleGraphs

Rigid

GraphData[{"Prism",15},"Rigid"]

GraphData[{"Prism",15},"Graph","Rigid"]//StyleGraphs

Flexible

GraphData[{"Prism",15},"Flexible"]

GraphData[{"Prism",15},"Graph","Flexible"]//StyleGraphs

LCF

GraphData[{"Prism",15},"LCF"]

GraphData[{"Prism",15},"HamiltonianCycleCount"]

GraphData[{"Prism",15},"LCFSignature"]

GraphData[{"Prism",15},"Graph","LCF"]//StyleGraphs

Tally[LCFNotation[#][[2]]&/@%]

LCF compute

Length[FindHamiltonianCycle[GraphData[{"Prism",15}],All][[All,All,1]]]//Timing

Tally[(lcf=LCF[GraphData[{"Prism",15}],All])[[All,2]]]//Timing

LCFGraph@@@lcf//StyleGraphs

Tally[LCFNotation[#][[-1]]&/@(gs=NiceLCFGraphs[lcf,2,1])]//Timing

Length[gs]

gs

UnitDistance

GraphData[{"Prism",15},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"Prism",15}],Debug->True]//Timing

GraphData[{"Prism",15},"Graph","UnitDistance"]//StyleGraphs

Select[%,!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&]

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData[{"Prism",15},"VertexList"],GraphData[{"Prism",15},"EdgeList"]]]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]]]&/@Complement@@(GraphData[{"Prism",15},"Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData[{"Prism",15}],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData[{"Prism",15}],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance by FindSubgraphUnitDistanceEmbedding

{{"Prism",15},l=FindSubgraphUnitDistanceEmbedding[GraphData[{"Prism",15}],TimeConstraint->300,"Results"->All]//Timing}

gs=SortBy[UniqueEmbeddings[l],Replace[RectilinearCrossingCount[#],Infinity->0]&]

gs={,,};

Matchstick

GraphData[{"Prism",15},"Matchstick"]

GraphData[{"Prism",15},"Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData[{"Prism",15},"Edges"]]

3D All

GraphData[{"Prism",15},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"Prism",15},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"Prism",15},"Polyhedral"]

GraphData[{"Prism",15},"Graph",{"3D","Polyhedron"}]

Graphics3D[{FaceForm[Green,Red],GraphData[{"Prism",15},"Polyhedron"]}]

3D Canonical

GraphData[{"Prism",15},"Graph",{"3D","Canonical"}]

Graphics3D[{Opacity[.5],Red,GraphData[{"Prism",15},"Polyhedron","Canonical"],Yellow,Sphere[]}]

{"Prism", 16}

Names

GraphData[{"Prism",16},"StandardName"]

GraphData[{"Prism",16},"StandardNames"]

Primary

GraphData[{"Prism",16}]//StyleGraphs[#,ImageSize->Small]&

Read and Wilson

All

GraphData[{"Prism",16},"Graph","All"]//StyleGraphs

Inexact

Select[GraphData[{"Prism",16},"Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData[{"Prism",16},"EdgeList"]]

Default embedding

GraphPlot@GraphData[{"Prism",16},"Edges"]

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

Degenerate

GraphData[{"Prism",16},"Graph","Degenerate"]//StyleGraphs

GeneralizedHoneycombGraph construction

With[{n=32},Cases[Table[{k,g=GeneralizedHoneycombGraph[n,k],RecognizeGraph@g},{k,n-1}],{__,{"Prism",_}}]]//Timing

g=GeneralizedHoneycombGraph[32,11]

RecognizeGraph@g

Bilateral

GraphData[{"Prism",16},"Graph","Bilateral"]//StyleGraphs

Circular

GraphData[{"Prism",16},"Graph","Circular"]//StyleGraphs

Planar

GraphData[{"Prism",16},"Planar"]

GraphData[{"Prism",16},"Graph","Planar"]//StyleGraphs

Tutte embeddings

faces=ReverseSort@SymmetricallyDistinctFaces[GraphData[{"Prism",16}]]

Tally[ReverseSort[Length/@faces]]

IGLayoutTutte[GraphData[{"Prism",16}],"OuterFace"->#]&/@faces

gs=Annotate[#,{VertexLabels->None,ImageSize->Tiny}]&/@{}

RecognizeGraph/@gs

MinimalCrossing

GraphData[{"Prism",16},{"RCN","CN"}]

GraphData[{"Prism",16},"Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

(cn=CrossingNumber[GraphData[{"Prism",16}],TimeConstraint->10])//AbsoluteTiming

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData[{"Prism",16}],"CrossingNumber"->cn,RandomSeed->n]],{n,50}],Max[OuterFace[#]]<=GraphData[{"Prism",16},"V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData[{"Prism",16}],"CrossingNumber"->cn,RandomSeed->i],{i,50}]//Union

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData[{"Prism",16},"Circulant"]

GraphData[{"Prism",16},"Graph","Circulant"]//StyleGraphs

Halin

GraphData[{"Prism",16},"Halin"]

GraphData[{"Prism",16},"Graph","Halin"]//StyleGraphs

Rigid

GraphData[{"Prism",16},"Rigid"]

GraphData[{"Prism",16},"Graph","Rigid"]//StyleGraphs

Flexible

GraphData[{"Prism",16},"Flexible"]

GraphData[{"Prism",16},"Graph","Flexible"]//StyleGraphs

LCF

GraphData[{"Prism",16},"LCF"]

GraphData[{"Prism",16},"HamiltonianCycleCount"]

GraphData[{"Prism",16},"LCFSignature"]

GraphData[{"Prism",16},"Graph","LCF"]//StyleGraphs

Tally[LCFNotation[#][[2]]&/@%]

LCF compute

Length[FindHamiltonianCycle[GraphData[{"Prism",16}],All][[All,All,1]]]//Timing

Tally[(lcf=LCF[GraphData[{"Prism",16}],All])[[All,2]]]//Timing

LCFGraph@@@lcf//StyleGraphs

Tally[LCFNotation[#][[-1]]&/@(gs=NiceLCFGraphs[lcf,2,1])]//Timing

Length[gs]

gs

UnitDistance

GraphData[{"Prism",16},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"Prism",16}],Debug->True]//Timing

GraphData[{"Prism",16},"Graph","UnitDistance"]//StyleGraphs

Select[%,!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&]

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData[{"Prism",16},"VertexList"],GraphData[{"Prism",16},"EdgeList"]]]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]]]&/@Complement@@(GraphData[{"Prism",16},"Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData[{"Prism",16}],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData[{"Prism",16}],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance by FindSubgraphUnitDistanceEmbedding

{{"Prism",16},l=FindSubgraphUnitDistanceEmbedding[GraphData[{"Prism",16}],TimeConstraint->300,"Results"->All]//Timing}

gs=SortBy[UniqueEmbeddings[l],Replace[RectilinearCrossingCount[#],Infinity->0]&]

gs={,,,,,,,,,,,,,,,,,,,,,};

Matchstick

GraphData[{"Prism",16},"Matchstick"]

GraphData[{"Prism",16},"Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData[{"Prism",16},"Edges"]]

3D All

GraphData[{"Prism",16},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"Prism",16},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"Prism",16},"Polyhedral"]

GraphData[{"Prism",16},"Graph",{"3D","Polyhedron"}]

Graphics3D[{FaceForm[Green,Red],GraphData[{"Prism",16},"Polyhedron"]}]

3D Canonical

GraphData[{"Prism",16},"Graph",{"3D","Canonical"}]

Graphics3D[{Opacity[.5],Red,GraphData[{"Prism",16},"Polyhedron","Canonical"],Yellow,Sphere[]}]

{"Prism", 17}

Names

GraphData[{"Prism",17},"StandardName"]

GraphData[{"Prism",17},"StandardNames"]

Primary

GraphData[{"Prism",17}]//StyleGraphs[#,ImageSize->Small]&

All

GraphData[{"Prism",17},"Graph","All"]//StyleGraphs

Inexact

Select[GraphData[{"Prism",17},"Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData[{"Prism",17},"EdgeList"]]

Default embedding

GraphPlot@GraphData[{"Prism",17},"Edges"]

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

Degenerate

GraphData[{"Prism",17},"Graph","Degenerate"]//StyleGraphs

Bilateral

GraphData[{"Prism",17},"Graph","Bilateral"]//StyleGraphs

Circular

GraphData[{"Prism",17},"Graph","Circular"]//StyleGraphs

Planar

GraphData[{"Prism",17},"Planar"]

GraphData[{"Prism",17},"Graph","Planar"]//StyleGraphs

Tutte embeddings

faces=ReverseSort@SymmetricallyDistinctFaces[GraphData[{"Prism",17}]]

Tally[ReverseSort[Length/@faces]]

IGLayoutTutte[GraphData[{"Prism",17}],"OuterFace"->#]&/@faces

gs=Annotate[#,{VertexLabels->None,ImageSize->Tiny}]&/@{}

RecognizeGraph/@gs

MinimalCrossing

GraphData[{"Prism",17},{"RCN","CN"}]

GraphData[{"Prism",17},"Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

(cn=CrossingNumber[GraphData[{"Prism",17}],TimeConstraint->10])//AbsoluteTiming

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData[{"Prism",17}],"CrossingNumber"->cn,RandomSeed->n]],{n,50}],Max[OuterFace[#]]<=GraphData[{"Prism",17},"V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData[{"Prism",17}],"CrossingNumber"->cn,RandomSeed->i],{i,50}]//Union

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData[{"Prism",17},"Circulant"]

GraphData[{"Prism",17},"Graph","Circulant"]//StyleGraphs

Halin

GraphData[{"Prism",17},"Halin"]

GraphData[{"Prism",17},"Graph","Halin"]//StyleGraphs

Rigid

GraphData[{"Prism",17},"Rigid"]

GraphData[{"Prism",17},"Graph","Rigid"]//StyleGraphs

Flexible

GraphData[{"Prism",17},"Flexible"]

GraphData[{"Prism",17},"Graph","Flexible"]//StyleGraphs

LCF

GraphData[{"Prism",17},"LCF"]

GraphData[{"Prism",17},"HamiltonianCycleCount"]

GraphData[{"Prism",17},"LCFSignature"]

GraphData[{"Prism",17},"Graph","LCF"]//StyleGraphs

Tally[LCFNotation[#][[2]]&/@%]

LCF compute

Length[FindHamiltonianCycle[GraphData[{"Prism",17}],All][[All,All,1]]]//Timing

Tally[(lcf=LCF[GraphData[{"Prism",17}],All])[[All,2]]]//Timing

lcf

Tally[LCFNotation[#][[-1]]&/@(gs=NiceLCFGraphs[lcf,2,1])]//Timing

Length[gs]

gs

UnitDistance

GraphData[{"Prism",17},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"Prism",17}],Debug->True]//Timing

GraphData[{"Prism",17},"Graph","UnitDistance"]//StyleGraphs

Select[%,!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&]

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData[{"Prism",17},"VertexList"],GraphData[{"Prism",17},"EdgeList"]]]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]]]&/@Complement@@(GraphData[{"Prism",17},"Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData[{"Prism",17}],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData[{"Prism",17}],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance by FindSubgraphUnitDistanceEmbedding

{{"Prism",17},l=FindSubgraphUnitDistanceEmbedding[GraphData[{"Prism",17}],TimeConstraint->300,"Results"->All]//Timing}

gs=SortBy[UniqueEmbeddings[l],Replace[RectilinearCrossingCount[#],Infinity->0]&]

g=;

Matchstick

GraphData[{"Prism",17},"Matchstick"]

GraphData[{"Prism",17},"Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData[{"Prism",17},"Edges"]]

3D All

GraphData[{"Prism",17},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"Prism",17},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"Prism",17},"Polyhedral"]

GraphData[{"Prism",17},"Graph",{"3D","Polyhedron"}]

Graphics3D[{FaceForm[Green,Red],GraphData[{"Prism",17},"Polyhedron"]}]

3D Canonical

GraphData[{"Prism",17},"Graph",{"3D","Canonical"}]

Graphics3D[{Opacity[.5],Red,GraphData[{"Prism",17},"Polyhedron","Canonical"],Yellow,Sphere[]}]

{"Prism", 18}

Names

GraphData[{"Prism",18},"StandardName"]

GraphData[{"Prism",18},"StandardNames"]

Primary

GraphData[{"Prism",18}]//StyleGraphs[#,ImageSize->Small]&

All

GraphData[{"Prism",18},"Graph","All"]//StyleGraphs

Inexact

Select[GraphData[{"Prism",18},"Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData[{"Prism",18},"EdgeList"]]

Default embedding

GraphPlot@GraphData[{"Prism",18},"Edges"]

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

Degenerate

GraphData[{"Prism",18},"Graph","Degenerate"]//StyleGraphs

Bilateral

GraphData[{"Prism",18},"Graph","Bilateral"]//StyleGraphs

Circular

GraphData[{"Prism",18},"Graph","Circular"]//StyleGraphs

Planar

GraphData[{"Prism",18},"Planar"]

GraphData[{"Prism",18},"Graph","Planar"]//StyleGraphs

Tutte embeddings

faces=ReverseSort@SymmetricallyDistinctFaces[GraphData[{"Prism",18}]]

Tally[ReverseSort[Length/@faces]]

IGLayoutTutte[GraphData[{"Prism",18}],"OuterFace"->#]&/@faces

gs=Annotate[#,{VertexLabels->None,ImageSize->Tiny}]&/@{}

RecognizeGraph/@gs

Tutte embedding construction

GraphPlot[RotateGraph[]]

v=Join[];
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

MinimalCrossing

GraphData[{"Prism",18},{"RCN","CN"}]

GraphData[{"Prism",18},"Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

(cn=CrossingNumber[GraphData[{"Prism",18}],TimeConstraint->10])//AbsoluteTiming

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData[{"Prism",18}],"CrossingNumber"->cn,RandomSeed->n]],{n,50}],Max[OuterFace[#]]<=GraphData[{"Prism",18},"V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData[{"Prism",18}],"CrossingNumber"->cn,RandomSeed->i],{i,50}]//Union

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData[{"Prism",18},"Circulant"]

GraphData[{"Prism",18},"Graph","Circulant"]//StyleGraphs

Halin

GraphData[{"Prism",18},"Halin"]

GraphData[{"Prism",18},"Graph","Halin"]//StyleGraphs

Rigid

GraphData[{"Prism",18},"Rigid"]

GraphData[{"Prism",18},"Graph","Rigid"]//StyleGraphs

Flexible

GraphData[{"Prism",18},"Flexible"]

GraphData[{"Prism",18},"Graph","Flexible"]//StyleGraphs

LCF

GraphData[{"Prism",18},"LCF"]

GraphData[{"Prism",18},"HamiltonianCycleCount"]

GraphData[{"Prism",18},"LCFSignature"]

GraphData[{"Prism",18},"Graph","LCF"]//StyleGraphs

Tally[LCFNotation[#][[2]]&/@%]

LCF compute

Length[FindHamiltonianCycle[GraphData[{"Prism",18}],All][[All,All,1]]]//Timing

Tally[(lcf=LCF[GraphData[{"Prism",18}],All])[[All,2]]]//Timing

Tally[LCFNotation[#][[-1]]&/@(gs=NiceLCFGraphs[lcf,2,1])]//Timing

Length[gs]

gs

UnitDistance

GraphData[{"Prism",18},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"Prism",18}],Debug->True]//Timing

GraphData[{"Prism",18},"Graph","UnitDistance"]//StyleGraphs

Select[%,!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&]

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData[{"Prism",18},"VertexList"],GraphData[{"Prism",18},"EdgeList"]]]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]]]&/@Complement@@(GraphData[{"Prism",18},"Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData[{"Prism",18}],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData[{"Prism",18}],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance by FindSubgraphUnitDistanceEmbedding

{{"Prism",18},l=FindSubgraphUnitDistanceEmbedding[GraphData[{"Prism",18}],TimeConstraint->300,"Results"->All]//Timing}

gs=SortBy[UniqueEmbeddings[l],Replace[RectilinearCrossingCount[#],Infinity->0]&]

gs={,,,,,,,,,,,,,,,,,};

Matchstick

GraphData[{"Prism",18},"Matchstick"]

GraphData[{"Prism",18},"Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData[{"Prism",18},"Edges"]]

3D All

GraphData[{"Prism",18},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"Prism",18},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"Prism",18},"Polyhedral"]

GraphData[{"Prism",18},"Graph",{"3D","Polyhedron"}]

Graphics3D[{FaceForm[Green,Red],GraphData[{"Prism",18},"Polyhedron"]}]

3D Canonical

GraphData[{"Prism",18},"Graph",{"3D","Canonical"}]

Graphics3D[{Opacity[.5],Red,GraphData[{"Prism",18},"Polyhedron","Canonical"],Yellow,Sphere[]}]

{"Prism", 19}

Names

GraphData[{"Prism",19},"StandardName"]

GraphData[{"Prism",19},"StandardNames"]

Primary

GraphData[{"Prism",19}]//StyleGraphs[#,ImageSize->Small]&

All

GraphData[{"Prism",19},"Graph","All"]//StyleGraphs

Inexact

Select[GraphData[{"Prism",19},"Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData[{"Prism",19},"EdgeList"]]

Default embedding

GraphPlot@GraphData[{"Prism",19},"Edges"]

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

Degenerate

GraphData[{"Prism",19},"Graph","Degenerate"]//StyleGraphs

Bilateral

GraphData[{"Prism",19},"Graph","Bilateral"]//StyleGraphs

Circular

GraphData[{"Prism",19},"Graph","Circular"]//StyleGraphs

Planar

GraphData[{"Prism",19},"Planar"]

GraphData[{"Prism",19},"Graph","Planar"]//StyleGraphs

Tutte embeddings

faces=ReverseSort@SymmetricallyDistinctFaces[GraphData[{"Prism",19}]]

Tally[ReverseSort[Length/@faces]]

IGLayoutTutte[GraphData[{"Prism",19}],"OuterFace"->#]&/@faces

gs=Annotate[#,{VertexLabels->None,ImageSize->Tiny}]&/@{}

RecognizeGraph/@gs

Tutte embedding construction

GraphPlot[RotateGraph[]]

v=Join[];
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

MinimalCrossing

GraphData[{"Prism",19},{"RCN","CN"}]

GraphData[{"Prism",19},"Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

(cn=CrossingNumber[GraphData[{"Prism",19}],TimeConstraint->10])//AbsoluteTiming

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData[{"Prism",19}],"CrossingNumber"->cn,RandomSeed->n]],{n,50}],Max[OuterFace[#]]<=GraphData[{"Prism",19},"V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData[{"Prism",19}],"CrossingNumber"->cn,RandomSeed->i],{i,50}]//Union

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData[{"Prism",19},"Circulant"]

GraphData[{"Prism",19},"Graph","Circulant"]//StyleGraphs

Halin

GraphData[{"Prism",19},"Halin"]

GraphData[{"Prism",19},"Graph","Halin"]//StyleGraphs

Rigid

GraphData[{"Prism",19},"Rigid"]

GraphData[{"Prism",19},"Graph","Rigid"]//StyleGraphs

Flexible

GraphData[{"Prism",19},"Flexible"]

GraphData[{"Prism",19},"Graph","Flexible"]//StyleGraphs

LCF

GraphData[{"Prism",19},"LCF"]

GraphData[{"Prism",19},"HamiltonianCycleCount"]

GraphData[{"Prism",19},"LCFSignature"]

GraphData[{"Prism",19},"Graph","LCF"]//StyleGraphs

Tally[LCFNotation[#][[2]]&/@%]

LCF compute

Length[FindHamiltonianCycle[GraphData[{"Prism",19}],All][[All,All,1]]]//Timing

Tally[(lcf=LCF[GraphData[{"Prism",19}],All])[[All,2]]]//Timing

Tally[LCFNotation[#][[-1]]&/@(gs=NiceLCFGraphs[lcf,2,1])]//Timing

Length[gs]

gs

UnitDistance

GraphData[{"Prism",19},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"Prism",19}],Debug->True]//Timing

GraphData[{"Prism",19},"Graph","UnitDistance"]//StyleGraphs

Select[%,!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&]

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData[{"Prism",19},"VertexList"],GraphData[{"Prism",19},"EdgeList"]]]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]]]&/@Complement@@(GraphData[{"Prism",19},"Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData[{"Prism",19}],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData[{"Prism",19}],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance by FindSubgraphUnitDistanceEmbedding

{{"Prism",19},l=FindSubgraphUnitDistanceEmbedding[GraphData[{"Prism",19}],TimeConstraint->300,"Results"->All]//Timing}

gs=SortBy[UniqueEmbeddings[l],Replace[RectilinearCrossingCount[#],Infinity->0]&]

gs={};

Matchstick

GraphData[{"Prism",19},"Matchstick"]

GraphData[{"Prism",19},"Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData[{"Prism",19},"Edges"]]

3D All

GraphData[{"Prism",19},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"Prism",19},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"Prism",19},"Polyhedral"]

GraphData[{"Prism",19},"Graph",{"3D","Polyhedron"}]

Graphics3D[{FaceForm[Green,Red],GraphData[{"Prism",19},"Polyhedron"]}]

3D Canonical

GraphData[{"Prism",19},"Graph",{"3D","Canonical"}]

Graphics3D[{Opacity[.5],Red,GraphData[{"Prism",19},"Polyhedron","Canonical"],Yellow,Sphere[]}]

{"Prism", 20}

Names

GraphData[{"Prism",20},"StandardName"]

GraphData[{"Prism",20},"StandardNames"]

Primary

GraphData[{"Prism",20}]//StyleGraphs[#,ImageSize->Small]&

All

GraphData[{"Prism",20},"Graph","All"]//StyleGraphs

Inexact

Select[GraphData[{"Prism",20},"Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData[{"Prism",20},"EdgeList"]]

Default embedding

GraphPlot@GraphData[{"Prism",20},"Edges"]

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

Degenerate

GraphData[{"Prism",20},"Graph","Degenerate"]//StyleGraphs

GeneralizedHoneycombGraph construction

With[{n=40},Cases[Table[{k,g=GeneralizedHoneycombGraph[n,k],RecognizeGraph@g},{k,n-1}],{__,{"Prism",_}}]]//Timing

g=GeneralizedHoneycombGraph[40,13]

RecognizeGraph@g

Bilateral

GraphData[{"Prism",20},"Graph","Bilateral"]//StyleGraphs

Circular

GraphData[{"Prism",20},"Graph","Circular"]//StyleGraphs

Planar

GraphData[{"Prism",20},"Planar"]

GraphData[{"Prism",20},"Graph","Planar"]//StyleGraphs

Tutte embeddings

faces=ReverseSort@SymmetricallyDistinctFaces[GraphData[{"Prism",20}]]

Tally[ReverseSort[Length/@faces]]

IGLayoutTutte[GraphData[{"Prism",20}],"OuterFace"->#]&/@faces

gs=Annotate[#,{VertexLabels->None,ImageSize->Tiny}]&/@{}

RecognizeGraph/@gs

MinimalCrossing

GraphData[{"Prism",20},{"RCN","CN"}]

GraphData[{"Prism",20},"Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

(cn=CrossingNumber[GraphData[{"Prism",20}],TimeConstraint->10])//AbsoluteTiming

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData[{"Prism",20}],"CrossingNumber"->cn,RandomSeed->n]],{n,50}],Max[OuterFace[#]]<=GraphData[{"Prism",20},"V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData[{"Prism",20}],"CrossingNumber"->cn,RandomSeed->i],{i,50}]//Union

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData[{"Prism",20},"Circulant"]

GraphData[{"Prism",20},"Graph","Circulant"]//StyleGraphs

Halin

GraphData[{"Prism",20},"Halin"]

GraphData[{"Prism",20},"Graph","Halin"]//StyleGraphs

Rigid

GraphData[{"Prism",20},"Rigid"]

GraphData[{"Prism",20},"Graph","Rigid"]//StyleGraphs

Flexible

GraphData[{"Prism",20},"Flexible"]

GraphData[{"Prism",20},"Graph","Flexible"]//StyleGraphs

LCF

GraphData[{"Prism",20},"LCF"]

GraphData[{"Prism",20},"HamiltonianCycleCount"]

GraphData[{"Prism",20},"LCFSignature"]

GraphData[{"Prism",20},"Graph","LCF"]//StyleGraphs

Tally[LCFNotation[#][[2]]&/@%]

LCF compute

Length[FindHamiltonianCycle[GraphData[{"Prism",20}],All][[All,All,1]]]//Timing

Tally[(lcf=LCF[GraphData[{"Prism",20}],All])[[All,2]]]//Timing

LCFGraph@@@lcf//StyleGraphs

Tally[LCFNotation[#][[-1]]&/@(gs=NiceLCFGraphs[lcf,2,1])]//Timing

Length[gs]

gs

UnitDistance

GraphData[{"Prism",20},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"Prism",20}],Debug->True]//Timing

GraphData[{"Prism",20},"Graph","UnitDistance"]//StyleGraphs

Select[%,!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&]

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData[{"Prism",20},"VertexList"],GraphData[{"Prism",20},"EdgeList"]]]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]]]&/@Complement@@(GraphData[{"Prism",20},"Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData[{"Prism",20}],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData[{"Prism",20}],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance by FindSubgraphUnitDistanceEmbedding

{{"Prism",20},l=FindSubgraphUnitDistanceEmbedding[GraphData[{"Prism",20}],TimeConstraint->300,"Results"->All]//Timing}

gs=SortBy[UniqueEmbeddings[l],Replace[RectilinearCrossingCount[#],Infinity->0]&]

gs={,,,,,,,,,,,,,,,,,}

Matchstick

GraphData[{"Prism",20},"Matchstick"]

GraphData[{"Prism",20},"Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData[{"Prism",20},"Edges"]]

3D All

GraphData[{"Prism",20},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"Prism",20},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"Prism",20},"Polyhedral"]

GraphData[{"Prism",20},"Graph",{"3D","Polyhedron"}]

Graphics3D[{FaceForm[Green,Red],GraphData[{"Prism",20},"Polyhedron"]}]

3D Canonical

GraphData[{"Prism",20},"Graph",{"3D","Canonical"}]

Graphics3D[{Opacity[.5],Red,GraphData[{"Prism",20},"Polyhedron","Canonical"],Yellow,Sphere[]}]

{"Prism", 21}

Names

GraphData[{"Prism",21},"StandardName"]

GraphData[{"Prism",21},"StandardNames"]

Primary

GraphData[{"Prism",21}]//StyleGraphs[#,ImageSize->Small]&

All

GraphData[{"Prism",21},"Graph","All"]//StyleGraphs

Inexact

Select[GraphData[{"Prism",21},"Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData[{"Prism",21},"EdgeList"]]

Default embedding

GraphPlot@GraphData[{"Prism",21},"Edges"]

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

Degenerate

GraphData[{"Prism",21},"Graph","Degenerate"]//StyleGraphs

Bilateral

GraphData[{"Prism",21},"Graph","Bilateral"]//StyleGraphs

Circular

GraphData[{"Prism",21},"Graph","Circular"]//StyleGraphs

Planar

GraphData[{"Prism",21},"Planar"]

GraphData[{"Prism",21},"Graph","Planar"]//StyleGraphs

Tutte embeddings

faces=ReverseSort@SymmetricallyDistinctFaces[GraphData[{"Prism",21}]]

Tally[ReverseSort[Length/@faces]]

IGLayoutTutte[GraphData[{"Prism",21}],"OuterFace"->#]&/@faces

gs=Annotate[#,{VertexLabels->None,ImageSize->Tiny}]&/@{}

RecognizeGraph/@gs

Tutte embedding construction

GraphPlot[RotateGraph[]]

v=Join[];
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

MinimalCrossing

GraphData[{"Prism",21},{"RCN","CN"}]

GraphData[{"Prism",21},"Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

(cn=CrossingNumber[GraphData[{"Prism",21}],TimeConstraint->10])//AbsoluteTiming

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData[{"Prism",21}],"CrossingNumber"->cn,RandomSeed->n]],{n,50}],Max[OuterFace[#]]<=GraphData[{"Prism",21},"V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData[{"Prism",21}],"CrossingNumber"->cn,RandomSeed->i],{i,50}]//Union

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData[{"Prism",21},"Circulant"]

GraphData[{"Prism",21},"Graph","Circulant"]//StyleGraphs

Halin

GraphData[{"Prism",21},"Halin"]

GraphData[{"Prism",21},"Graph","Halin"]//StyleGraphs

Rigid

GraphData[{"Prism",21},"Rigid"]

GraphData[{"Prism",21},"Graph","Rigid"]//StyleGraphs

Flexible

GraphData[{"Prism",21},"Flexible"]

GraphData[{"Prism",21},"Graph","Flexible"]//StyleGraphs

LCF

GraphData[{"Prism",21},"LCF"]

GraphData[{"Prism",21},"HamiltonianCycleCount"]

GraphData[{"Prism",21},"LCFSignature"]

GraphData[{"Prism",21},"Graph","LCF"]//StyleGraphs

Tally[LCFNotation[#][[2]]&/@%]

LCF compute

Length[FindHamiltonianCycle[GraphData[{"Prism",21}],All][[All,All,1]]]//Timing

Tally[(lcf=LCF[GraphData[{"Prism",21}],All])[[All,2]]]//Timing

LCFGraph@@@lcf//StyleGraphs

Tally[LCFNotation[#][[-1]]&/@(gs=NiceLCFGraphs[lcf,2,1])]//Timing

Length[gs]

UnitDistance

GraphData[{"Prism",21},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"Prism",21}],Debug->True]//Timing

GraphData[{"Prism",21},"Graph","UnitDistance"]//StyleGraphs

Select[%,!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&]

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData[{"Prism",21},"VertexList"],GraphData[{"Prism",21},"EdgeList"]]]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]]]&/@Complement@@(GraphData[{"Prism",21},"Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData[{"Prism",21}],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData[{"Prism",21}],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance by FindSubgraphUnitDistanceEmbedding

{{"Prism",21},l=FindSubgraphUnitDistanceEmbedding[GraphData[{"Prism",21}],TimeConstraint->300,"Results"->All]//Timing}

gs=SortBy[UniqueEmbeddings[l],Replace[RectilinearCrossingCount[#],Infinity->0]&]

Matchstick

GraphData[{"Prism",21},"Matchstick"]

GraphData[{"Prism",21},"Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData[{"Prism",21},"Edges"]]

3D All

GraphData[{"Prism",21},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"Prism",21},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"Prism",21},"Polyhedral"]

GraphData[{"Prism",21},"Graph",{"3D","Polyhedron"}]

Graphics3D[{FaceForm[Green,Red],GraphData[{"Prism",21},"Polyhedron"]}]

3D Canonical

GraphData[{"Prism",21},"Graph",{"3D","Canonical"}]

Graphics3D[{Opacity[.5],Red,GraphData[{"Prism",21},"Polyhedron","Canonical"],Yellow,Sphere[]}]

{"Prism", 22}

Names

GraphData[{"Prism",22},"StandardName"]

GraphData[{"Prism",22},"StandardNames"]

Primary

GraphData[{"Prism",22}]//StyleGraphs[#,ImageSize->Small]&

All

GraphData[{"Prism",22},"Graph","All"]//StyleGraphs

Inexact

Select[GraphData[{"Prism",22},"Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData[{"Prism",22},"EdgeList"]]

Default embedding

GraphPlot@GraphData[{"Prism",22},"Edges"]

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

Degenerate

GraphData[{"Prism",22},"Graph","Degenerate"]//StyleGraphs

DegenerateGraphEmbeddingTypes/@GraphData[{"Prism",22},"Graph","Degenerate"]

Perspective

GraphData[{"Prism",22},"Graph","Perspective"]//StyleGraphs

Perspective construction

PolyhedronGraph[GraphData[{"Prism",22},"Polyhedron"]]

Manipulate[PolyhedronProjectionGraph[GraphData[{"Prism",22},"Polyhedron"],{x,y,z}],{x,-1,1},{y,-1,1},{z,-1,1}]

PolyhedronProjectionGraph[GraphData[{"Prism",22},"Polyhedron"],{0,.01,.01}]

PolyhedronProjectionGraph[GraphData[{"Prism",22},"Polyhedron"],{0,0,1}]

PolyhedronProjectionGraph[GraphData[{"Prism",22},"Polyhedron"],{0,1,0}]

PolyhedronProjectionGraph[GraphData[{"Prism",22},"Polyhedron"],{1,0.01,0}]

PolyhedronProjectionGraph[GraphData[{"Prism",22},"Polyhedron"],{1,1,0}]

PolyhedronProjectionGraph[GraphData[{"Prism",22},"Polyhedron"],{1,0,1,-Pi/2}]

PolyhedronProjectionGraph[GraphData[{"Prism",22},"Polyhedron"],{1,1,1}]

GeneralizedHoneycombGraph construction

With[{n=44},Cases[Table[{k,g=GeneralizedHoneycombGraph[n,k],RecognizeGraph@g},{k,n-1}],{__,{"Prism",_}}]]//Timing

g=GeneralizedHoneycombGraph[44,15]

RecognizeGraph@g

Bilateral

GraphData[{"Prism",22},"Graph","Bilateral"]//StyleGraphs

Circular

GraphData[{"Prism",22},"Graph","Circular"]//StyleGraphs

Planar

GraphData[{"Prism",22},"Planar"]

GraphData[{"Prism",22},"Graph","Planar"]//StyleGraphs

Tutte embeddings

faces=ReverseSort@SymmetricallyDistinctFaces[GraphData[{"Prism",22}]]

Tally[ReverseSort[Length/@faces]]

IGLayoutTutte[GraphData[{"Prism",22}],"OuterFace"->#]&/@faces

gs=Annotate[#,{VertexLabels->None,ImageSize->Tiny}]&/@{}

RecognizeGraph/@gs

MinimalCrossing

GraphData[{"Prism",22},{"RCN","CN"}]

GraphData[{"Prism",22},"Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

(cn=CrossingNumber[GraphData[{"Prism",22}],TimeConstraint->1])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"Prism",22}],TimeConstraint->10])//AbsoluteTiming

(g=MinimalCrossingGraph[GraphData[{"Prism",22}],"CrossingNumber"->cn])//PolyhedralGraphQ

faces=ReverseSort@SymmetricallyDistinctFaces[g]

Tally[ReverseSort[Length/@faces]]

IGLayoutTutte[g,"OuterFace"->#]&/@faces

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData[{"Prism",22}],"CrossingNumber"->cn,RandomSeed->n]],{n,50}],Max[OuterFace[#]]<=GraphData[{"Prism",22},"V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData[{"Prism",22}],"CrossingNumber"->cn,RandomSeed->i],{i,50}]//Union

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData[{"Prism",22},"Circulant"]

GraphData[{"Prism",22},"Graph","Circulant"]//StyleGraphs

Complement graph

GraphData[{"Prism",22},"ComplementGraph","Name"]

Line graph

GraphData[{"Prism",22},"Line"]

LineGraphQ[GraphData[{"Prism",22}]]//Timing

RootGraph

Select[GraphData[All],GraphData[#,"LineGraph","Name"]=== {"Prism",22}&]

RootGraph[GraphData[{"Prism",22}]]

Halin

GraphData[{"Prism",22},"Halin"]

GraphData[{"Prism",22},"Graph","Halin"]//StyleGraphs

Rigid

GraphData[{"Prism",22},"Rigid"]

GraphData[{"Prism",22},"Graph","Rigid"]//StyleGraphs

Flexible

GraphData[{"Prism",22},"Flexible"]

GraphData[{"Prism",22},"Graph","Flexible"]//StyleGraphs

LCF

GraphData[{"Prism",22},"LCF"]

GraphData[{"Prism",22},"HamiltonianCycleCount"]

GraphData[{"Prism",22},"LCFSignature"]

GraphData[{"Prism",22},"LCFNotations"]

GraphData[{"Prism",22},"Graph","LCF"]//StyleGraphs

Tally[LCFNotation[#][[2]]&/@%]

LCF compute all

Divisors[GraphData[{"Prism",22},"V"]]

Length[FindHamiltonianCycle[GraphData[{"Prism",22}],All][[All,All,1]]]//Timing

Tally[(lcf=LCF[GraphData[{"Prism",22}],All])[[All,2]]]//Timing

lcf

LCFGraph@@@DeleteCases[lcf,{_,1}]//StyleGraphs

Tally[LCFNotation[#][[-1]]&/@(gs=NiceLCFGraphs[lcf,2,1])]//Timing

Length[gs]

gs

UnitDistance

GraphData[{"Prism",22},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"Prism",22}],Debug->True]//Timing

GraphData[{"Prism",22},"Graph","UnitDistance"]//StyleGraphs

Select[%,!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&]

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData[{"Prism",22},"VertexList"],GraphData[{"Prism",22},"EdgeList"]],MaxIterations->500]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]],MaxIterations->500]&/@Complement@@(GraphData[{"Prism",22},"Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData[{"Prism",22}],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData[{"Prism",22}],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance by FindSubgraphUnitDistanceEmbedding

{{"Prism",22},l=FindSubgraphUnitDistanceEmbedding[GraphData[{"Prism",22}],TimeConstraint->300,"Results"->All]//Timing}

gs=SortBy[UniqueEmbeddings[l],Replace[RectilinearCrossingCount[#],Infinity->0]&]

gs={,,,,,,,,,,};

Matchstick

GraphData[{"Prism",22},"Matchstick"]

GraphData[{"Prism",22},"Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData[{"Prism",22},"Edges"]]

3D All

GraphData[{"Prism",22},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"Prism",22},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"Prism",22},"Polyhedral"]

GraphData[{"Prism",22},"Graph",{"3D","Polyhedron"}]

Graphics3D[{FaceForm[Green,Red],GraphData[{"Prism",22},"Polyhedron"]}]

{"Prism", 23}

Names

GraphData[{"Prism",23},"StandardName"]

GraphData[{"Prism",23},"StandardNames"]

Primary

GraphData[{"Prism",23}]//StyleGraphs[#,ImageSize->Small]&

All

GraphData[{"Prism",23},"Graph","All"]//StyleGraphs

Inexact

Select[GraphData[{"Prism",23},"Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData[{"Prism",23},"EdgeList"]]

Default embedding

GraphPlot@GraphData[{"Prism",23},"Edges"]

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

Degenerate

GraphData[{"Prism",23},"Graph","Degenerate"]//StyleGraphs

Bilateral

GraphData[{"Prism",23},"Graph","Bilateral"]//StyleGraphs

Circular

GraphData[{"Prism",23},"Graph","Circular"]//StyleGraphs

Planar

GraphData[{"Prism",23},"Planar"]

GraphData[{"Prism",23},"Graph","Planar"]//StyleGraphs

Tutte embeddings

faces=ReverseSort@SymmetricallyDistinctFaces[GraphData[{"Prism",23}]]

Tally[ReverseSort[Length/@faces]]

IGLayoutTutte[GraphData[{"Prism",23}],"OuterFace"->#]&/@faces

gs=Annotate[#,{VertexLabels->None,ImageSize->Tiny}]&/@{}

RecognizeGraph/@gs

Tutte embedding construction

GraphPlot[RotateGraph[]]

v=Join[];
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

MinimalCrossing

GraphData[{"Prism",23},{"RCN","CN"}]

GraphData[{"Prism",23},"Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

(cn=CrossingNumber[GraphData[{"Prism",23}],TimeConstraint->10])//AbsoluteTiming

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData[{"Prism",23}],"CrossingNumber"->cn,RandomSeed->n]],{n,50}],Max[OuterFace[#]]<=GraphData[{"Prism",23},"V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData[{"Prism",23}],"CrossingNumber"->cn,RandomSeed->i],{i,50}]//Union

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData[{"Prism",23},"Circulant"]

GraphData[{"Prism",23},"Graph","Circulant"]//StyleGraphs

Halin

GraphData[{"Prism",23},"Halin"]

GraphData[{"Prism",23},"Graph","Halin"]//StyleGraphs

Rigid

GraphData[{"Prism",23},"Rigid"]

GraphData[{"Prism",23},"Graph","Rigid"]//StyleGraphs

Flexible

GraphData[{"Prism",23},"Flexible"]

GraphData[{"Prism",23},"Graph","Flexible"]//StyleGraphs

LCF

GraphData[{"Prism",23},"LCF"]

GraphData[{"Prism",23},"HamiltonianCycleCount"]

GraphData[{"Prism",23},"LCFSignature"]

GraphData[{"Prism",23},"Graph","LCF"]//StyleGraphs

Tally[LCFNotation[#][[2]]&/@%]

LCF compute

Length[FindHamiltonianCycle[GraphData[{"Prism",23}],All][[All,All,1]]]//Timing

Tally[(lcf=LCF[GraphData[{"Prism",23}],All])[[All,2]]]//Timing

LCFGraph@@@lcf//StyleGraphs

Tally[LCFNotation[#][[-1]]&/@(gs=NiceLCFGraphs[lcf,2,1])]//Timing

Length[gs]

UnitDistance

GraphData[{"Prism",23},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"Prism",23}],Debug->True]//Timing

GraphData[{"Prism",23},"Graph","UnitDistance"]//StyleGraphs

Select[%,!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&]

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData[{"Prism",23},"VertexList"],GraphData[{"Prism",23},"EdgeList"]]]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]]]&/@Complement@@(GraphData[{"Prism",23},"Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData[{"Prism",23}],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData[{"Prism",23}],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance by FindSubgraphUnitDistanceEmbedding

{{"Prism",23},l=FindSubgraphUnitDistanceEmbedding[GraphData[{"Prism",23}],TimeConstraint->300,"Results"->All]//Timing}

gs=SortBy[UniqueEmbeddings[l],Replace[RectilinearCrossingCount[#],Infinity->0]&]

Matchstick

GraphData[{"Prism",23},"Matchstick"]

GraphData[{"Prism",23},"Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData[{"Prism",23},"Edges"]]

3D All

GraphData[{"Prism",23},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"Prism",23},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"Prism",23},"Polyhedral"]

GraphData[{"Prism",23},"Graph",{"3D","Polyhedron"}]

Graphics3D[{FaceForm[Green,Red],GraphData[{"Prism",23},"Polyhedron"]}]

3D Canonical

GraphData[{"Prism",23},"Graph",{"3D","Canonical"}]

Graphics3D[{Opacity[.5],Red,GraphData[{"Prism",23},"Polyhedron","Canonical"],Yellow,Sphere[]}]

{"Prism", 24}

Names

GraphData[{"Prism",24},"StandardName"]

GraphData[{"Prism",24},"StandardNames"]

Primary

GraphData[{"Prism",24}]//StyleGraphs[#,ImageSize->Small]&

All

GraphData[{"Prism",24},"Graph","All"]//StyleGraphs

Inexact

Select[GraphData[{"Prism",24},"Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData[{"Prism",24},"EdgeList"]]

Default embedding

GraphPlot@GraphData[{"Prism",24},"Edges"]

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

Degenerate

GraphData[{"Prism",24},"Graph","Degenerate"]//StyleGraphs

DegenerateGraphEmbeddingTypes/@GraphData[{"Prism",24},"Graph","Degenerate"]

Perspective

GraphData[{"Prism",24},"Graph","Perspective"]//StyleGraphs

Perspective construction

PolyhedronGraph[GraphData[{"Prism",24},"Polyhedron"]]

Manipulate[PolyhedronProjectionGraph[GraphData[{"Prism",24},"Polyhedron"],{x,y,z}],{x,-1,1},{y,-1,1},{z,-1,1}]

PolyhedronProjectionGraph[GraphData[{"Prism",24},"Polyhedron"],{0,.01,.01}]

PolyhedronProjectionGraph[GraphData[{"Prism",24},"Polyhedron"],{0,0,1}]

PolyhedronProjectionGraph[GraphData[{"Prism",24},"Polyhedron"],{0,1,0}]

PolyhedronProjectionGraph[GraphData[{"Prism",24},"Polyhedron"],{1,0.01,0}]

PolyhedronProjectionGraph[GraphData[{"Prism",24},"Polyhedron"],{1,1,0}]

PolyhedronProjectionGraph[GraphData[{"Prism",24},"Polyhedron"],{1,0,1,-Pi/2}]

PolyhedronProjectionGraph[GraphData[{"Prism",24},"Polyhedron"],{1,1,1}]

Bilateral

GraphData[{"Prism",24},"Graph","Bilateral"]//StyleGraphs

Circular

GraphData[{"Prism",24},"Graph","Circular"]//StyleGraphs

Planar

GraphData[{"Prism",24},"Planar"]

GraphData[{"Prism",24},"Graph","Planar"]//StyleGraphs

Tutte embeddings

faces=ReverseSort@SymmetricallyDistinctFaces[GraphData[{"Prism",24}]]

Tally[ReverseSort[Length/@faces]]

IGLayoutTutte[GraphData[{"Prism",24}],"OuterFace"->#]&/@faces

gs=Annotate[#,{VertexLabels->None,ImageSize->Tiny}]&/@{}

RecognizeGraph/@gs

MinimalCrossing

GraphData[{"Prism",24},{"RCN","CN"}]

GraphData[{"Prism",24},"Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

(cn=CrossingNumber[GraphData[{"Prism",24}],TimeConstraint->1])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"Prism",24}],TimeConstraint->10])//AbsoluteTiming

(g=MinimalCrossingGraph[GraphData[{"Prism",24}],"CrossingNumber"->cn])//PolyhedralGraphQ

faces=ReverseSort@SymmetricallyDistinctFaces[g]

Tally[ReverseSort[Length/@faces]]

IGLayoutTutte[g,"OuterFace"->#]&/@faces

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData[{"Prism",24}],"CrossingNumber"->cn,RandomSeed->n]],{n,50}],Max[OuterFace[#]]<=GraphData[{"Prism",24},"V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData[{"Prism",24}],"CrossingNumber"->cn,RandomSeed->i],{i,50}]//Union

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData[{"Prism",24},"Circulant"]

GraphData[{"Prism",24},"Graph","Circulant"]//StyleGraphs

Complement graph

GraphData[{"Prism",24},"ComplementGraph","Name"]

Line graph

GraphData[{"Prism",24},"Line"]

LineGraphQ[GraphData[{"Prism",24}]]//Timing

RootGraph

Select[GraphData[All],GraphData[#,"LineGraph","Name"]=== {"Prism",24}&]

RootGraph[GraphData[{"Prism",24}]]

Halin

GraphData[{"Prism",24},"Halin"]

GraphData[{"Prism",24},"Graph","Halin"]//StyleGraphs

Rigid

GraphData[{"Prism",24},"Rigid"]

GraphData[{"Prism",24},"Graph","Rigid"]//StyleGraphs

Flexible

GraphData[{"Prism",24},"Flexible"]

GraphData[{"Prism",24},"Graph","Flexible"]//StyleGraphs

LCF

GraphData[{"Prism",24},"LCF"]

GraphData[{"Prism",24},"HamiltonianCycleCount"]

GraphData[{"Prism",24},"LCFSignature"]

GraphData[{"Prism",24},"LCFNotations"]

GraphData[{"Prism",24},"Graph","LCF"]//StyleGraphs

Tally[LCFNotation[#][[2]]&/@%]

LCF compute all

Divisors[GraphData[{"Prism",24},"V"]]

Length[FindHamiltonianCycle[GraphData[{"Prism",24}],All][[All,All,1]]]//Timing

Tally[(lcf=LCF[GraphData[{"Prism",24}],All])[[All,2]]]//Timing

DeleteCases[lcf,{_,1}]

LCFGraph@@@DeleteCases[lcf,{_,1}]//StyleGraphs

Tally[LCFNotation[#][[-1]]&/@(gs=NiceLCFGraphs[lcf,2,1])]//Timing

Length[gs]

gs

UnitDistance

GraphData[{"Prism",24},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"Prism",24}],Debug->True]//Timing

GraphData[{"Prism",24},"Graph","UnitDistance"]//StyleGraphs

Select[%,!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&]

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData[{"Prism",24},"VertexList"],GraphData[{"Prism",24},"EdgeList"]],MaxIterations->500]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]],MaxIterations->500]&/@Complement@@(GraphData[{"Prism",24},"Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData[{"Prism",24}],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData[{"Prism",24}],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance by FindSubgraphUnitDistanceEmbedding

{{"Prism",24},l=FindSubgraphUnitDistanceEmbedding[GraphData[{"Prism",24}],TimeConstraint->300,"Results"->All]//Timing}

gs=SortBy[UniqueEmbeddings[l],Replace[RectilinearCrossingCount[#],Infinity->0]&]

gs={,,,,,,,,};

Matchstick

GraphData[{"Prism",24},"Matchstick"]

GraphData[{"Prism",24},"Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData[{"Prism",24},"Edges"]]

3D All

GraphData[{"Prism",24},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"Prism",24},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"Prism",24},"Polyhedral"]

GraphData[{"Prism",24},"Graph",{"3D","Polyhedron"}]

3D Canonical

GraphData[{"Prism",24},"Graph",{"3D","Canonical"}]

Graphics3D[{Opacity[.5],Red,GraphData[{"Prism",24},"Polyhedron","Canonical"],Yellow,Sphere[]}]

{"Prism", 25}

Names

GraphData[{"Prism",25},"StandardName"]

GraphData[{"Prism",25},"StandardNames"]

Primary

GraphData[{"Prism",25}]//StyleGraphs[#,ImageSize->Small]&

All

GraphData[{"Prism",25},"Graph","All"]//StyleGraphs

Inexact

Select[GraphData[{"Prism",25},"Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData[{"Prism",25},"EdgeList"]]

Default embedding

GraphPlot@GraphData[{"Prism",25},"Edges"]

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

Degenerate

GraphData[{"Prism",25},"Graph","Degenerate"]//StyleGraphs

DegenerateGraphEmbeddingTypes/@GraphData[{"Prism",25},"Graph","Degenerate"]

Perspective

GraphData[{"Prism",25},"Graph","Perspective"]//StyleGraphs

Perspective construction

PolyhedronGraph[GraphData[{"Prism",25},"Polyhedron"]]

Manipulate[PolyhedronProjectionGraph[GraphData[{"Prism",25},"Polyhedron"],{x,y,z}],{x,-1,1},{y,-1,1},{z,-1,1}]

PolyhedronProjectionGraph[GraphData[{"Prism",25},"Polyhedron"],{0,.01,.01}]

PolyhedronProjectionGraph[GraphData[{"Prism",25},"Polyhedron"],{0,0,1}]

PolyhedronProjectionGraph[GraphData[{"Prism",25},"Polyhedron"],{0,1,0}]

PolyhedronProjectionGraph[GraphData[{"Prism",25},"Polyhedron"],{1,0.01,0}]

PolyhedronProjectionGraph[GraphData[{"Prism",25},"Polyhedron"],{1,1,0}]

PolyhedronProjectionGraph[GraphData[{"Prism",25},"Polyhedron"],{1,0,1,-Pi/2}]

PolyhedronProjectionGraph[GraphData[{"Prism",25},"Polyhedron"],{1,1,1}]

Bilateral

GraphData[{"Prism",25},"Graph","Bilateral"]//StyleGraphs

Circular

GraphData[{"Prism",25},"Graph","Circular"]//StyleGraphs

Planar

GraphData[{"Prism",25},"Planar"]

GraphData[{"Prism",25},"Graph","Planar"]//StyleGraphs

Tutte embeddings

faces=ReverseSort@SymmetricallyDistinctFaces[GraphData[{"Prism",25}]]

Tally[ReverseSort[Length/@faces]]

IGLayoutTutte[GraphData[{"Prism",25}],"OuterFace"->#]&/@faces

gs=Annotate[#,{VertexLabels->None,ImageSize->Tiny}]&/@{}

RecognizeGraph/@gs

MinimalCrossing

GraphData[{"Prism",25},{"RCN","CN"}]

GraphData[{"Prism",25},"Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

(cn=CrossingNumber[GraphData[{"Prism",25}],TimeConstraint->1])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"Prism",25}],TimeConstraint->10])//AbsoluteTiming

(g=MinimalCrossingGraph[GraphData[{"Prism",25}],"CrossingNumber"->cn])//PolyhedralGraphQ

faces=ReverseSort@SymmetricallyDistinctFaces[g]

Tally[ReverseSort[Length/@faces]]

IGLayoutTutte[g,"OuterFace"->#]&/@faces

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData[{"Prism",25}],"CrossingNumber"->cn,RandomSeed->n]],{n,50}],Max[OuterFace[#]]<=GraphData[{"Prism",25},"V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData[{"Prism",25}],"CrossingNumber"->cn,RandomSeed->i],{i,50}]//Union

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData[{"Prism",25},"Circulant"]

GraphData[{"Prism",25},"Graph","Circulant"]//StyleGraphs

Complement graph

GraphData[{"Prism",25},"ComplementGraph","Name"]

Line graph

GraphData[{"Prism",25},"Line"]

LineGraphQ[GraphData[{"Prism",25}]]//Timing

RootGraph

Select[GraphData[All],GraphData[#,"LineGraph","Name"]=== {"Prism",25}&]

RootGraph[GraphData[{"Prism",25}]]

Halin

GraphData[{"Prism",25},"Halin"]

GraphData[{"Prism",25},"Graph","Halin"]//StyleGraphs

Rigid

GraphData[{"Prism",25},"Rigid"]

GraphData[{"Prism",25},"Graph","Rigid"]//StyleGraphs

Flexible

GraphData[{"Prism",25},"Flexible"]

GraphData[{"Prism",25},"Graph","Flexible"]//StyleGraphs

LCF

GraphData[{"Prism",25},"LCF"]

GraphData[{"Prism",25},"HamiltonianCycleCount"]

GraphData[{"Prism",25},"LCFSignature"]

GraphData[{"Prism",25},"LCFNotations"]

GraphData[{"Prism",25},"Graph","LCF"]//StyleGraphs

Tally[LCFNotation[#][[2]]&/@%]

LCF compute all

Divisors[GraphData[{"Prism",25},"V"]]

Length[FindHamiltonianCycle[GraphData[{"Prism",25}],All][[All,All,1]]]//Timing

Tally[(lcf=LCF[GraphData[{"Prism",25}],All])[[All,2]]]//Timing

lcf

LCFGraph@@@DeleteCases[lcf,{_,1}]//StyleGraphs

Tally[LCFNotation[#][[-1]]&/@(gs=NiceLCFGraphs[lcf,2,1])]//Timing

Length[gs]

gs

UnitDistance

GraphData[{"Prism",25},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"Prism",25}],Debug->True]//Timing

GraphData[{"Prism",25},"Graph","UnitDistance"]//StyleGraphs

Select[%,!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&]

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData[{"Prism",25},"VertexList"],GraphData[{"Prism",25},"EdgeList"]],MaxIterations->500]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]],MaxIterations->500]&/@Complement@@(GraphData[{"Prism",25},"Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData[{"Prism",25}],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData[{"Prism",25}],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance by FindSubgraphUnitDistanceEmbedding

{{"Prism",25},l=FindSubgraphUnitDistanceEmbedding[GraphData[{"Prism",25}],TimeConstraint->300,"Results"->All]//Timing}

gs=SortBy[UniqueEmbeddings[l],Replace[RectilinearCrossingCount[#],Infinity->0]&]

Matchstick

GraphData[{"Prism",25},"Matchstick"]

GraphData[{"Prism",25},"Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData[{"Prism",25},"Edges"]]

3D All

GraphData[{"Prism",25},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"Prism",25},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"Prism",25},"Polyhedral"]

GraphData[{"Prism",25},"Graph",{"3D","Polyhedron"}]

Graphics3D[{FaceForm[Green,Red],GraphData[{"Prism",25},"Polyhedron"]}]

3D Canonical

GraphData[{"Prism",25},"Graph",{"3D","Canonical"}]

Graphics3D[{Opacity[.5],Red,GraphData[{"Prism",25},"Polyhedron","Canonical"],Yellow,Sphere[]}]

Properties

Acyclic

Table[GraphData[{"Prism",n},"Acyclic"],{n,3,25}]

AdjacencyMatrixCount

seq=Table[GraphData[{"Prism",n},"AdjacencyMatrixCount"],{n,3,25}]

FindSequenceFunction[seq,n-2]

(x^3)FindGeneratingFunction[seq,x]

Table[(((2n)!)/(Piecewise[{{48,n==4}},4 n])),{n,3,8}]

FullSimplify[(((2n)!)/(Piecewise[{{48,n==4}},4 n])),n>2]

Table[(1/2) Gamma[2 n],{n,3,8}]

Arboricity

Table[GraphData[{"Prism",n},"Arboricity"],{n,3,25}]

ArcTransitive

Table[GraphData[{"Prism",n},"ArcTransitive"],{n,3,25}]

ArcTransitivity

Table[GraphData[{"Prism",n},"ArcTransitivity"],{n,3,25}]

AutomorphismCount

A124351

seq=Table[GraphData[{"Prism",n},"AutomorphismCount"],{n,3,25}]

FindSequenceFunction[Drop[seq,2],n-4]

Table[4n,{n,3,25}]

BalabanIndex

seq=Table[GraphData[{"Prism",n},"BalabanIndex"],{n,3,25}]

seq=Table[BalabanIndex[PrismGraph[n]],{n,25}]

FindLinearRecurrence[seq]

FullSimplify[FindSequenceFunction[seq,n]]

Table[((9 (((-1))^n) n (-1+(((-1))^n) (-1+2 n (2+n))))/((((2+n))^2) (-1+n (2+n)))),{n,25}]

Factor/@FullSimplify[((9 (((-1))^n) n (-1+(((-1))^n) (-1+2 n (2+n))))/((((2+n))^2) (-1+n (2+n)))),Mod[n,2]==#]&/@{0,1}

Table[Piecewise[{{((18 n)/(((2+n))^2)),Mod[n,2]==0},{((18 (n^2))/((2+n) (-1+2 n+(n^2)))),Mod[n,2]==1}}],{n,25}]

gf=(x^3)FindGeneratingFunction[seq,x]

Series[gf,{x,0,10}]//Simplify

Print[{#,BalabanIndex[GraphData[#]]//Timing}]&/@Table[{"Prism",n},{n,21,25,2}];

Bandwidth

Table[GraphData[{"Prism",n},"Bandwidth"],{n,3,25}]

BridgeCount

seq=Table[GraphData[{"Prism",n},"BridgeCount"],{n,3,25}]

BurningNumber

seq=Table[GraphData[{"Prism",n},"BurningNumber"],{n,3,25}]

ChordCount

seq=Table[GraphData[{"Prism",n},"ChordCount"],{n,3,20}]

Table[GraphData[{"Prism",n},"EdgeCount"],{n,3,20}]

FindSequenceFunction[%,n-2]//Simplify

Chordless

seq=Table[GraphData[{"Prism",n},"Chordless"],{n,3,20}]

ChordlessCycleCount

seq=Table[GraphData[{"Prism",n},"ChordlessCycleCount"],{n,3,25}]

FindSequenceFunction[Rest@seq,n-3]//FullSimplify

gf=(x^3)FindGeneratingFunction[seq,x]

FullSimplify[SeriesCoefficient[gf,{x,0,n}],n>2]

Table[2 Cos[((n π)/3)],{n,10}]

Table[((((1/2) (1-(Sqrt[5]))))^n)+((((1/2) (1+(Sqrt[5]))))^n),{n,10}]//FullSimplify

Table[LucasL[n],{n,10}]//FullSimplify

Table[LucasL[n]+n+2 Cos[((n π)/3)],{n,3,25}]

Table[Piecewise[{{3,n==3}},LucasL[n]+n+2 Cos[((n π)/3)]],{n,3,25}]

ChromaticInvariant

seq=Table[GraphData[{"Prism",n},"ChromaticInvariant"],{n,3,25}]

FindSequenceFunction[Drop[seq,-5],n-2]

ChromaticNumber

Entity["Graph",{"Prism",n}]["ChromaticNumber"]

seq=Table[GraphData[{"Prism",n},"ChromaticNumber"],{n,3,25}]

FindSequenceFunction[seq,n-2]//FullSimplify

Table[Piecewise[{{2,Mod[n,2]==0},{3,Mod[n,2]==1}}],{n,3,25}]

ChromaticPolynomial

Entity["Graph",{"Prism",n}]["ChromaticPolynomial"][x]

(poly=Factor@Table[GraphData[{"Prism",n},"ChromaticPolynomial"],{n,k0=3,kmax=25}]/.f_Function:>f[x])//Column

CoefficientList[Take[poly,5],x]

coef=Factor/@FindLinearRecurrence[Take[DeleteMissing[poly],15]]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

FullSimplify[SeriesCoefficient[gf,{z,0,n}],n>2]

poly2=Table[XXX,{n,20}]

FullSimplify[poly2-poly]

Circumference

seq=Table[GraphData[{"Prism",n},"Circumference"],{n,3,25}]

FindSequenceFunction[seq,n-2]

Classes

Intersection@@(GraphData[#,"Classes"]&/@GraphData["Prism"])

CliqueCount

seq=Table[GraphData[{"Prism",n},"CliqueCount"],{n,3,25}]

FindSequenceFunction[seq,n-2]

gf=(x^3)FindGeneratingFunction[seq,x]

FullSimplify[SeriesCoefficient[gf,{x,0,n}],n>2]

Table[Piecewise[{{18, n == 3}, {1 + 5*n, n > 3}}], {n, 3, 25}]

CliqueCoveringNumber

seq=Table[GraphData[{"Prism",n},"CliqueCoveringNumber"],{n,3,25}]

FindSequenceFunction[Rest@seq,n-3]//FullSimplify

CliqueNumber

seq=Table[GraphData[{"Prism",n},"CliqueNumber"],{n,3,25}]

ComplementOddChordlessCycleCount

seq=Table[GraphData[{"Prism",n},"ComplementOddChordlessCycleCount"],{n,3,25}]

ConnectedDominatingSetCount

A286985

seq=Table[GraphData[{"Prism",n},"ConnectedDominatingSetCount"],{n,3,25}]

seq={7,7,39,115,343,967,2663,7203,19239,50887,133543,348179,902775,2329607,5986535,15327555,39115847,99532423,252601127,639548595,1615746455,4073951559,10253517671,25763632995,64635943783,161928486727,405134009511,1012371656275};

coef=FindLinearRecurrence[seq]

Length@coef

Take[seq,6]

LinearRecurrence[{6,-11,4,5,-2,-1},{7,7,39,115,343,967},20]

FullSimplify[FindSequenceFunction[seq,n],n∈Integers]

Table[(((1-(Sqrt[2])))^n)+(((1+(Sqrt[2])))^n),{n,10}]//Expand

Table[LucasL[n,2],{n,10}]

Table[((((1+(Sqrt[2])))^n) (-5+4 (Sqrt[2]))-(((1-(Sqrt[2])))^n) (5+4 (Sqrt[2]))),{n,10}]//Expand

%14/2

Table[3Fibonacci[n-2,2]+Fibonacci[n-1,2],{n,10}]

Table[1+LucasL[n,2]+n(-2+2(3Fibonacci[n-2,2]+Fibonacci[n-1,2])),{n,20}]//Expand

FullSimplify[1+LucasL[n,2]+n(-2+2(3Fibonacci[n-2,2]+Fibonacci[n-1,2]))]

Table[LucasL[n,2]+2n(3Fibonacci[n-2,2]+Fibonacci[n-1,2]-1)+1,{n,20}]

gf=x FindGeneratingFunction[seq,x]

CoefficientList[Series[((7-35 x+74 (x^2)-70 (x^3)+19 (x^4)-3 (x^5))/(((1-3 x+(x^2)+(x^3)))^2)),{x,0,20}],x]

ConnectedDominationNumber

seq=Table[GraphData[{"Prism",n},"ConnectedDominationNumber"],{n,3,25}]

ConnectedDominationPolynomial

(poly=Factor[Table[GraphData[{"Prism",n},"ConnectedDominationPolynomial"],{n,k0=3,kmax=20}]/.f_Function:>f[x]])//Column

CoefficientList[Take[poly,5],x]

coef=Factor/@FindLinearRecurrence[DeleteMissing@poly]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

FullSimplify[SeriesCoefficient[gf,{z,0,n}],n>0]

poly2=Table[XXX,{n,20}]

FullSimplify[poly2-poly]

ConnectedInducedSubgraphCount

A286182

seq=Table[GraphData[{"Prism",n},"ConnectedInducedSubgraphCount"],{n,3,25}]

seq={3,13,51,167,503,1441,4007,10923,29355,78037,205659,538127,1399583,3621289,9327695,23931603,61186131};

FindLinearRecurrence[DeleteMissing[seq]]

gf=x FindGeneratingFunction[DeleteMissing[seq],x]

FullSimplify[SeriesCoefficient[gf,{x,0,n}],n>0]

Table[1+(((1-(Sqrt[2])))^n)+(((1+(Sqrt[2])))^n)-(3/4) (4+(Sqrt[2]) (((1-(Sqrt[2])))^n)-(Sqrt[2]) (((1+(Sqrt[2])))^n)) n,{n,10}]//Expand

Table[1+(1/4)((((1-(Sqrt[2])))^n)(4-3(Sqrt[2])n)+(((1+(Sqrt[2])))^n)(4+3(Sqrt[2])n))-3n,{n,10}]//Expand

Table[LucasL[n,2]+3n Fibonacci[n,2]-3n+1,{n,20}]

ConnectedInducedSubgraphPolynomial

(poly=Factor@Table[GraphData[{"Prism",n},"ConnectedInducedSubgraphPolynomial"],{n,k0=3,kmax=20}]/.f_Function:>f[x])//Column

CoefficientList[Take[poly,5],x]

coef=Factor/@FindLinearRecurrence[DeleteMissing[poly]]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

FullSimplify[SeriesCoefficient[gf,{z,0,n}],n>0]

poly2=Table[XXX,{n,20}]

FullSimplify[poly2-poly]

CrossingNumber

seq=Table[GraphData[{"Prism",n},"CrossingNumber"],{n,3,25}]

CycleCount

A077265

seq=Table[GraphData[{"Prism",n},"CycleCount"],{n,3,25}]

FindSequenceFunction[Take[seq,10],n-2]//FullSimplify

Limit[(((-(-1+x) x))^n)+(((x (1+x)))^n)+((n (-(x^4)+(x^(2+2 n))))/(-1+(x^2))),x->1,Assumptions->n>0]//FullSimplify

CyclePolynomial

(poly=Factor@(Table[GraphData[{"Prism",n},"CyclePolynomial"],{n,k0=3,kmax=15}]/.f_Function:>f[x]))//Column

coef=Factor/@FindLinearRecurrence[poly]

p=.

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

FullSimplify[SeriesCoefficient[gf,{z,0,n}],n>0]

poly2=Table[(((-(-1+x) x))^n)+(((x (1+x)))^n)+((n (-(x^4)+(x^(2+2 n))))/(-1+(x^2))),{n,k0,kmax}]//Factor

FullSimplify[poly2-poly]

poly3=Table[x^n ((1-x)^n+(1+x)^n)+(n (x^(2 (n+1))-x^4))/(x^2-1),{n,k0,kmax}];

FullSimplify[poly3-poly]

Cyclic

Table[GraphData[{"Prism",n},"Cyclic"],{n,3,25}]

CyclomaticNumber

seq=Table[GraphData[{"Prism",n},"CyclomaticNumber"],{n,3,25}]

FindSequenceFunction[seq,n-2]//FullSimplify

DetourIndex

seq=Table[GraphData[{"Prism",n},"DetourIndex"],{n,3,25}]

FindSequenceFunction[seq,n-2]//FullSimplify

gf=(x^3)FindGeneratingFunction[seq,x]

FullSimplify[SeriesCoefficient[gf,{x,0,n}],n>2]

Factor/@Simplify[(1/2) n (3+(((-1))^n)+n (-9+(((-1))^(1+n))+8 n)),Mod[n,2]==#]&/@{0,1}

Diameter

seq=Table[GraphData[{"Prism",n},"Diameter"],{n,3,25}]

Table[1+Floor[n/2],{n,3,25}]

DistinguishingNumber

seq=Table[GraphData[{"Prism",n},"DistinguishingNumber"],{n,3,25}]

DomaticNumber

Table[PrismGraph[n,VertexLabels->"Name"],{n,3,4}]

seq=Table[GraphData[{"Prism",n},"DomaticNumber"],{n,3,25}]

Table[Piecewise[{{4,Mod[n,4]==0}},3],{n,3,25}]

Do[Print[{n,DomaticNumber[PrismGraph[n]]//Timing}],{n,3,5}]

DominatingSetCount

A284702

seq=Table[GraphData[{"Prism",n},"DominatingSetCount"],{n,3,25}]

FindLinearRecurrence[DeleteMissing[seq]]

LinearRecurrence[{3,1,5,1,1,-1,-1},{3,11,51,183,663,2435,8935},20]

FindSequenceFunction[DeleteMissing[seq],n-2]

gf=x FindGeneratingFunction[DeleteMissing[seq],x]

bug(334115)

soln=Simplify[SeriesCoefficient[gf,{x,0,n}],n>0]

N[Drop[List@@soln,2]/.n->0]//Chop

FullSimplify[(((-I))^n)+(I^n),n∈Integers]

Table[2 Cos[((n π)/2)]+RootSum[1+#1-2 ((#1)^2)-2 ((#1)^3)-3 ((#1)^4)+((#1)^5)&,#^n&],{n,20}]

DominationNumber

seq=Table[GraphData[{"Prism",n},"DominationNumber"],{n,3,25}]

FindSequenceFunction[seq,n-2]//FullSimplify

Table[Floor[(n+2)/4]+Floor[(n+3)/4],{n,3,25}]

DominationPolynomial

(poly=Factor@(Table[GraphData[{"Prism",n},"DominationPolynomial"],{n,k0=3,kmax=20}]/.f_Function:>f[x]))//Column

CoefficientList[Take[poly,4],x]

coef=Factor/@FindLinearRecurrence[DeleteCases[poly,_Missing]]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

FullSimplify[SeriesCoefficient[gf,{z,0,n}],n>0]

Factor[Take[poly2-poly,kmax-k0-Count[poly,_Missing]+1]]

FullSimplify[poly2-poly]

EdgeChromaticNumber

seq=Table[GraphData[{"Prism",n},"EdgeChromaticNumber"],{n,3,25}]

EdgeConnectivity

Table[GraphData[{"Prism",n},"EdgeConnectivity"],{n,3,25}]

EdgeCount

seq=Table[GraphData[{"Prism",n},"EdgeCount"],{n,3,25}]

FindSequenceFunction[seq,n-2]

EdgeCoverCount

A123304

seq=Table[GraphData[{"Prism",n},"EdgeCoverCount"],{n,3,25}]

FindSequenceFunction[seq,n-2]

Table[(((-1))^(-2+n))+Root[-4+33 #1-42 ((#1)^2)+((#1)^3)&,2] ((Root[2-3 #1-6 ((#1)^2)+((#1)^3)&,1])^(-2+n))+Root[-4+33 #1-42 ((#1)^2)+((#1)^3)&,1] ((Root[2-3 #1-6 ((#1)^2)+((#1)^3)&,2])^(-2+n))+Root[-4+33 #1-42 ((#1)^2)+((#1)^3)&,3] ((Root[2-3 #1-6 ((#1)^2)+((#1)^3)&,3])^(-2+n)),{n,1,2}]//FullSimplify

FindSequenceFunction[{5,43,263,1699,10895,69943,448943,2881699,18497135,118730023,762108143,4891844659,31399932335,201550911703,1293721577903,8304182337859,53303156937455,342144045482503,2196165379031663,14096818096762579},n]

Table[(((-1))^n)+RootSum[2-3 #-6 ((#)^2)+((#)^3)&,#^n&],{n,3,25}]

%==seq

Table[(((-1))^n)+RootSum[2-3 #-6 ((#)^2)+((#)^3)&,#^n&],{n,0,20}]

EdgeCoverNumber

seq=Table[GraphData[{"Prism",n},"EdgeCoverNumber"],{n,3,25}]

FindSequenceFunction[seq,n-2]

EdgeCoverPolynomial

(poly=Factor[Table[GraphData[{"Prism",n},"EdgeCoverPolynomial"],{n,k0=3,kmax=25}]/.f_Function:>f[x]])//Column

CoefficientList[Take[poly,5],x]

coef=Factor/@FindLinearRecurrence[poly]

p=.

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

Simplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

Simplify[SeriesCoefficient[gf,{z,0,n}],n>0]

Factor[Take[poly2-poly,kmax-k0-Count[poly,_Missing]+1]]

FullSimplify[poly2-poly]

Eulerian

seq=Table[GraphData[{"Prism",n},"Eulerian"],{n,3,25}]

EulerianCycleCount

seq=Table[GraphData[{"Prism",n},"EulerianCycleCount"],{n,3,25}]

FaceCount

seq=Table[GraphData[{"Prism",n},"FaceCount"],{n,3,25}]

FibonacciCube

Entity["Graph",{"Prism",n}]["FibonacciCube"]

seq=Table[GraphData[{"Prism",n},"FibonacciCube"],{n,3,25}]

FractionalChromaticNumber

seq=Table[GraphData[{"Prism",n},"FractionalChromaticNumber"],{n,3,25}]

Table[Piecewise[{{2,Mod[n,2]==0},{((2n)/(n-1)),Mod[n,2]==1}}],{n,3,25}]

FractionalEdgeChromaticNumber

seq=Table[GraphData[{"Prism",n},"FractionalEdgeChromaticNumber"],{n,3,25}]

Girth

seq=Table[GraphData[{"Prism",n},"Girth"],{n,3,25}]

Gonality

seq=Table[GraphData[{"Prism",n},"Gonality"],{n,3,25}]

HamiltonConnected

seq=Table[GraphData[{"Prism",n},"HamiltonConnected"],{n,3,25}]

HamiltonDecompositionCount

seq=Table[GraphData[{"Prism",n},"HamiltonDecompositionCount"],{n,3,25}]

FullSimplify[FindSequenceFunction[seq,n-2],n∈Integers]

Hamiltonian

Table[GraphData[{"Prism",n},"Hamiltonian"],{n,3,25}]

HamiltonianCycleCount

1/2 * A124349

A014681(n) gives (n-1)-prism count

seq=Table[GraphData[{"Prism",n},"HamiltonianCycleCount"],{n,3,25}]

FullSimplify[FindSequenceFunction[seq,n-2],n∈Integers]

Table[1+(-1)^n+n,{n,3,25}]

HamiltonianPathCount

A124350

seq=Table[GraphData[{"Prism",n},"HamiltonianPathCount"],{n,3,25}]

FullSimplify[FindSequenceFunction[seq,n-2],n>2]

Table[n (2 n^2+(-1)^n+3)/2,{n,3,25}]

Table[2n (Floor[n^2/2]+1),{n,3,25}]

FindLinearRecurrence[seq]

HamiltonianWalkCount

seq=Table[GraphData[{"Prism",n},"HamiltonianWalkCount"],{n,3,25}]

HararyIndex

seq=Table[GraphData[{"Prism",n},"HararyIndex"],{n,3,25}]

seq=Table[HararyIndex[PrismGraph[n]],{n,3,30}]

FindLinearRecurrence[seq]

FindLinearRecurrence[Rest@seq]

FindSequenceFunction[seq,n-2]

FindSequenceFunction[Rest@seq,n-3]

Table[DifferenceRoot[Function[{\[FormalY],\[FormalN]},{-2424-1152 \[FormalN]-136 (\[FormalN]^2)+(-93-52 \[FormalN]-7 (\[FormalN]^2)) \[FormalY][\[FormalN]]+(-341-123 \[FormalN]-10 (\[FormalN]^2)) \[FormalY][1+\[FormalN]]+(-1+\[FormalN]) (39+7 \[FormalN]) \[FormalY][2+\[FormalN]]+(7+\[FormalN]) (39+10 \[FormalN]) \[FormalY][3+\[FormalN]]==0,\[FormalY][1]==12,\[FormalY][2]==(58/3),\[FormalY][3]==(85/3)}]][-2+n],{n,3,25}]

gf=(x^3)FindGeneratingFunction[seq,x]

HexagonCount

seq=Table[GraphData[{"Prism",n},"HexagonCount"],{n,3,25}]

FindSequenceFunction[seq,n-2]

gf=(x^3)FindGeneratingFunction[seq,x]

FullSimplify[SeriesCoefficient[gf,{x,0,n}],n>2]

HosoyaIndex

Same as IndependentEdgeSetCount

seq=Table[GraphData[{"Prism",n},"HosoyaIndex"],{n,3,25}]

FindSequenceFunction[seq,n-2]

Table[(((-1))^(-2+n))+Root[-1+7 #1-11 ((#1)^2)+((#1)^3)&,2] ((Root[1-#1-3 ((#1)^2)+((#1)^3)&,1])^(-2+n))+Root[-1+7 #1-11 ((#1)^2)+((#1)^3)&,1] ((Root[1-#1-3 ((#1)^2)+((#1)^3)&,2])^(-2+n))+Root[-1+7 #1-11 ((#1)^2)+((#1)^3)&,3] ((Root[1-#1-3 ((#1)^2)+((#1)^3)&,3])^(-2+n)),{n,1,2}]//FullSimplify

FindSequenceFunction[{2,12,32,108,342,1104,3544,11396,36626,117732,378424,1216380,3909830,12567448,40395792,129844996,417363330,1341539196,4312135920,13860583628},n]

Table[(-1)^n+RootSum[1-#-3 ((#)^2)+((#)^3)&,#^n&],{n,3,25}]

IdiosyncraticPolynomial

(poly=Table[GraphData[{"Prism",n},"IdiosyncraticPolynomial"],{n,k0=3,kmax=25}]/.f_Function:>f[x,y])//Column

2+Position[poly,_Missing]//Flatten

coef=Factor/@FindLinearRecurrence[Take[poly,15]]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

FullSimplify[SeriesCoefficient[gf,{z,0,n}],n>0]

Factor[Take[poly2-poly,kmax-k0-Count[poly,_Missing]+1]]

FullSimplify[poly2-poly]

IndependenceNumber

seq=Table[GraphData[{"Prism",n},"IndependenceNumber"],{n,3,25}]

Table[2 Floor[n/2],{n,3,25}]

IndependencePolynomial

(((-1))^n) (x^n) + A102413

(poly=Factor[Table[GraphData[{"Prism",n},"IndependencePolynomial"],{n,k0=3,kmax=25}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[DeleteMissing[poly]]

CoefficientList[Take[poly,5],x]

coef=Factor/@FindLinearRecurrence[Take[DeleteMissing[poly],10]]

p=.

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

CoefficientList[Table[(2^(-n))((((1+x-(Sqrt[1+x (6+x)])))^n)+(((1+x+(Sqrt[1+x (6+x)])))^n)),{n,10}]//Expand,x]

SeriesCoefficient[(1+t*z^2)/(1-(1+t)*z-t*z^2),{z,0,n}]

ExpandAll[ ((((1+t-(Sqrt[1+6 t+(t^2)])))^n)+(((1+t+(Sqrt[1+6 t+(t^2)])))^n))/.t->t-1]

Sum[

ExpandAll[ ((((1+t-(Sqrt[1+6 t+(t^2)])))^n)+(((1+t+(Sqrt[1+6 t+(t^2)])))^n))/.t->t+1]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

FullSimplify[SeriesCoefficient[gf,{z,0,n}],n>0]

poly2=Table[(((-1))^n)(x^n)+(2^(-n))((((1+x-(Sqrt[1+x (6+x)])))^n)+(((1+x+(Sqrt[1+x (6+x)])))^n)),{n,k0,kmax}];

FullSimplify[Take[poly2-poly,18]]

(1 + t*z^2) / (1 - (1+t)*z - t*z^2).

IndependenceRatio

seq=Table[GraphData[{"Prism",n},"IndependenceRatio"],{n,3,25}]

FindSequenceFunction[seq,n-2]//FullSimplify

Table[((Floor[(n/2)])/n),{n,3,25}]

IndependentEdgeSetCount

A102080

seq=Table[GraphData[{"Prism",n},"IndependentEdgeSetCount"],{n,3,25}]

FindLinearRecurrence[seq]

LinearRecurrence[{2,4,0,-1},{2,12,32,108},20]

FindSequenceFunction[seq,n-2]

Table[(((-1))^(-2+n))+Root[-1+7 #1-11 ((#1)^2)+((#1)^3)&,2] ((Root[1-#1-3 ((#1)^2)+((#1)^3)&,1])^(-2+n))+Root[-1+7 #1-11 ((#1)^2)+((#1)^3)&,1] ((Root[1-#1-3 ((#1)^2)+((#1)^3)&,2])^(-2+n))+Root[-1+7 #1-11 ((#1)^2)+((#1)^3)&,3] ((Root[1-#1-3 ((#1)^2)+((#1)^3)&,3])^(-2+n)),{n,1,2}]//FullSimplify

FindSequenceFunction[{2,12,32,108,342,1104,3544,11396,36626,117732,378424,1216380,3909830,12567448,40395792,129844996,417363330,1341539196,4312135920,13860583628},n]

seq=Table[(-1)^n+RootSum[1-#-3 ((#)^2)+((#)^3)&,#^n&],{n,25}]

gf=x FindGeneratingFunction[seq,x]

CoefficientList[Series[-((2 (-1-4 x+2 (x^3)))/(1-2 x-4 (x^2)+(x^4))),{x,0,20}],x]

IndependentVertexSetCount

A051927

seq=Table[GraphData[{"Prism",n},"IndependentVertexSetCount"],{n,3,25}]

FindSequenceFunction[seq,n-2]//FullSimplify

Table[LucasL[n,2]+(-1)^n,{n,3,25}]

IntersectionNumber

seq=Table[GraphData[{"Prism",n},"IntersectionNumber"],{n,3,25}]

IrredundantSetCount

A290511

Table[GraphData[{"Prism",n},"IrredundantSetCount"],{n,3,25}]

(seq=Table[Length[IrredundantSets[PrismGraph[n]]],{n,3,9}])//Timing

seq=Import["https://oeis.org/A290511/b290511.txt","Table"][[All,2]];

coef=FindLinearRecurrence[seq]

Length[coef]

Take[seq,Length[coef]]

LinearRecurrence[{3,0,-1,2,-6,-8,13,6,-6,-2,-5,-4,2,2,-1,1,3,0,2},{3,9,24,77,198,522,1550,4477,12732,36214,103579,296294,846303,2417368,6907329,19737901,56396602,161138214,460420182},{10}]

eqns=Prepend[Table[a[k]==seq[[k]],{k,Length[coef]}],a[n]==Sum[a[n-k]coef[[k]],{k,Length[coef]}]]

RSolve[eqns,a[n],n]

FindGeneratingFunction[Take[seq,40],x]//Head//Timing

FindGeneratingFunction[Take[seq,50],x]//Head//Timing

gf=x(3-3*x^2+8*x^3-30*x^4-48*x^5+91*x^6+48*x^7-54*x^8-20*x^9-55*x^10-48*x^11+26*x^12+28*x^13-15*x^14+16*x^15+51*x^16+38*x^18)/((1-x)*(1-x^2+x^4+x^6)*(1-2*x-x^2-3*x^3-5*x^4+2*x^5+6*x^6+5*x^7+4*x^8+4*x^9+3*x^10+2*x^11+2*x^12));

Series[gf,{x,0,20}]

res=Refine[SeriesCoefficient[gf,{x,0,n}],n>0]

part1=FullSimplify[RootReduce[1+((E^(-(1/2) I n π)) ((((-Root[1+#1-((#1)^2)+((#1)^3)&,1]))^(n/2)) ((-1+2 Root[1+#1-((#1)^2)+((#1)^3)&,1]-3 ((Root[1+#1-((#1)^2)+((#1)^3)&,1])^2)) Root[1+#1-((#1)^2)+((#1)^3)&,2]+(1-2 (1+(((-1))^n)) Root[1+#1-((#1)^2)+((#1)^3)&,1]+3 ((Root[1+#1-((#1)^2)+((#1)^3)&,1])^2)) Root[1+#1-((#1)^2)+((#1)^3)&,3])+(E^((I n π)/2)) (2 ((Root[1+#1-((#1)^2)+((#1)^3)&,1])^((2+n)/2)) Root[1+#1-((#1)^2)+((#1)^3)&,2]-3 ((Root[1+#1-((#1)^2)+((#1)^3)&,1])^((4+n)/2)) (Root[1+#1-((#1)^2)+((#1)^3)&,2]-Root[1+#1-((#1)^2)+((#1)^3)&,3])-(((-(Sqrt[Root[1+#1-((#1)^2)+((#1)^3)&,2]])))^n) Root[1+#1-((#1)^2)+((#1)^3)&,3]-3 (((-(Sqrt[Root[1+#1-((#1)^2)+((#1)^3)&,2]])))^n) ((Root[1+#1-((#1)^2)+((#1)^3)&,2])^2) Root[1+#1-((#1)^2)+((#1)^3)&,3]-((Root[1+#1-((#1)^2)+((#1)^3)&,2])^(n/2)) Root[1+#1-((#1)^2)+((#1)^3)&,3]+2 ((Root[1+#1-((#1)^2)+((#1)^3)&,2])^((2+n)/2)) Root[1+#1-((#1)^2)+((#1)^3)&,3]-3 ((Root[1+#1-((#1)^2)+((#1)^3)&,2])^((4+n)/2)) Root[1+#1-((#1)^2)+((#1)^3)&,3]+((Root[1+#1-((#1)^2)+((#1)^3)&,1])^(n/2)) (-Root[1+#1-((#1)^2)+((#1)^3)&,2]+Root[1+#1-((#1)^2)+((#1)^3)&,3])+Root[1+#1-((#1)^2)+((#1)^3)&,2] (2 (((-(Sqrt[Root[1+#1-((#1)^2)+((#1)^3)&,2]])))^n) Root[1+#1-((#1)^2)+((#1)^3)&,3]+(((-(Sqrt[Root[1+#1-((#1)^2)+((#1)^3)&,3]])))^n) (1-2 Root[1+#1-((#1)^2)+((#1)^3)&,3]+3 ((Root[1+#1-((#1)^2)+((#1)^3)&,3])^2))+((Root[1+#1-((#1)^2)+((#1)^3)&,3])^(n/2)) (1-2 Root[1+#1-((#1)^2)+((#1)^3)&,3]+3 ((Root[1+#1-((#1)^2)+((#1)^3)&,3])^2)))+Root[1+#1-((#1)^2)+((#1)^3)&,1] (((Root[1+#1-((#1)^2)+((#1)^3)&,2])^(n/2))-2 ((Root[1+#1-((#1)^2)+((#1)^3)&,2])^((2+n)/2))+3 ((Root[1+#1-((#1)^2)+((#1)^3)&,2])^((4+n)/2))+(((-(Sqrt[Root[1+#1-((#1)^2)+((#1)^3)&,2]])))^n) (1-2 Root[1+#1-((#1)^2)+((#1)^3)&,2]+3 ((Root[1+#1-((#1)^2)+((#1)^3)&,2])^2))-(1-2 Root[1+#1-((#1)^2)+((#1)^3)&,3]+3 ((Root[1+#1-((#1)^2)+((#1)^3)&,3])^2)) ((((-(Sqrt[Root[1+#1-((#1)^2)+((#1)^3)&,3]])))^n)+((Root[1+#1-((#1)^2)+((#1)^3)&,3])^(n/2)))))))/(2 (-2+Root[1+#1-((#1)^2)+((#1)^3)&,1] ((Root[1+#1-((#1)^2)+((#1)^3)&,2])^2)+((Root[1+#1-((#1)^2)+((#1)^3)&,1])^2) Root[1+#1-((#1)^2)+((#1)^3)&,3]+Root[1+#1-((#1)^2)+((#1)^3)&,2] ((Root[1+#1-((#1)^2)+((#1)^3)&,3])^2)))],n>0&&n∈Integers]

FullSimplify[RootReduce[Expand[%79]]/.((r_Root)^(n/2)):>((RootReduce[Sqrt[r]])^n),n∈Integers&&n>0]

Table[1+((Root[1+((#1)^2)-((#1)^4)+((#1)^6)&,1])^n)+((Root[1+((#1)^2)-((#1)^4)+((#1)^6)&,2])^n)+((Root[1+((#1)^2)-((#1)^4)+((#1)^6)&,5])^n)+((Root[1+((#1)^2)-((#1)^4)+((#1)^6)&,6])^n)+(I^(3 n)) ((Root[-1+((#1)^2)+((#1)^4)+((#1)^6)&,2])^n) (1+(((-1))^n) Root[1-11 ((#1)^2)+363 ((#1)^4)+1331 ((#1)^6)&,5])+((Root[1+((#1)^2)-((#1)^4)+((#1)^6)&,4])^n) (Root[-13+37 #1-33 ((#1)^2)+11 ((#1)^3)&,1]+Root[1-11 ((#1)^2)+363 ((#1)^4)+1331 ((#1)^6)&,6]),{n,10}]//Round

Table[1+RootSum[1+((#1)^2)-((#1)^4)+((#1)^6)&,#^n&],{n,10}]

terms=List@@((((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,1])^n) (2+6 Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,1]+12 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,1])^2)+16 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,1])^3)+25 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,1])^4)+36 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,1])^5)+14 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 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((#1)^11)+((#1)^12)&,1]-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,6]) (Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,1]-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,7]) (Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,1]-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,8]) (Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,1]-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,9]) (Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,1]-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,10]) (Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,1]-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,11]) (Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,1]-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,12]))+(((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,2])^n) (2+6 Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,2]+12 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,2])^2)+16 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,2])^3)+25 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,2])^4)+36 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,2])^5)+14 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,2])^6)-40 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,2])^7)-27 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,2])^8)-10 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,2])^9)-22 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,2])^10)+12 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,2])^11)))/((-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,1]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,2]) (Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,2]-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,3]) (Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,2]-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,4]) (Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,2]-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,5]) (Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,2]-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,6]) (Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,2]-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,7]) (Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,2]-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,8]) (Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,2]-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,9]) (Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,2]-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,10]) (Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,2]-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,11]) (Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,2]-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,12]))+(((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,3])^n) (2+6 Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,3]+12 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 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((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,3])^9)-22 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,3])^10)+12 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,3])^11)))/((-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,1]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,3]) (-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,2]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,3]) (Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,3]-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,4]) (Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,3]-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,5]) (Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,3]-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,6]) (Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,3]-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,7]) (Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,3]-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,8]) (Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,3]-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,9]) (Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,3]-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,10]) (Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,3]-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,11]) (Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,3]-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,12]))+(((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,4])^n) (2+6 Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,4]+12 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,4])^2)+16 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,4])^3)+25 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,4])^4)+36 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,4])^5)+14 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,4])^6)-40 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,4])^7)-27 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,4])^8)-10 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,4])^9)-22 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,4])^10)+12 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,4])^11)))/((-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,1]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,4]) (-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,2]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,4]) (-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,3]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,4]) (Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,4]-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,5]) (Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,4]-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,6]) (Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,4]-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,7]) (Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,4]-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,8]) (Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,4]-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,9]) (Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,4]-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,10]) (Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,4]-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,11]) (Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,4]-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,12]))+(((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,5])^n) (2+6 Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,5]+12 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,5])^2)+16 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,5])^3)+25 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,5])^4)+36 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,5])^5)+14 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,5])^6)-40 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,5])^7)-27 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,5])^8)-10 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,5])^9)-22 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,5])^10)+12 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,5])^11)))/((-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,1]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,5]) (-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,2]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,5]) (-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,3]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,5]) (-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,4]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,5]) (Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,5]-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,6]) (Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,5]-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,7]) (Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,5]-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,8]) (Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,5]-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,9]) (Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,5]-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,10]) (Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,5]-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,11]) (Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,5]-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,12]))+(((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,6])^n) (2+6 Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,6]+12 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,6])^2)+16 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,6])^3)+25 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,6])^4)+36 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,6])^5)+14 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,6])^6)-40 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,6])^7)-27 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,6])^8)-10 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,6])^9)-22 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,6])^10)+12 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,6])^11)))/((-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,1]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,6]) (-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,2]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,6]) (-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,3]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,6]) (-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,4]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,6]) (-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,5]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,6]) (Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,6]-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,7]) (Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,6]-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,8]) (Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,6]-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,9]) (Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,6]-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,10]) (Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,6]-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,11]) (Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,6]-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,12]))+(((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,7])^n) (2+6 Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,7]+12 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,7])^2)+16 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,7])^3)+25 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,7])^4)+36 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,7])^5)+14 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,7])^6)-40 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,7])^7)-27 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,7])^8)-10 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,7])^9)-22 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,7])^10)+12 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,7])^11)))/((-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,1]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,7]) (-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,2]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,7]) (-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,3]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,7]) (-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,4]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,7]) (-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,5]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,7]) (-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,6]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,7]) (Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,7]-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,8]) (Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,7]-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,9]) (Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,7]-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,10]) (Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,7]-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,11]) (Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,7]-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,12]))+(((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,8])^n) (2+6 Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,8]+12 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,8])^2)+16 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,8])^3)+25 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,8])^4)+36 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,8])^5)+14 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,8])^6)-40 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,8])^7)-27 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,8])^8)-10 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,8])^9)-22 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,8])^10)+12 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,8])^11)))/((-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,1]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,8]) (-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,2]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,8]) (-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,3]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,8]) (-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,4]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,8]) (-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,5]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,8]) (-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,6]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,8]) (-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,7]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,8]) (Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,8]-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,9]) (Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,8]-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,10]) (Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,8]-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,11]) (Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,8]-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,12]))+(((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,9])^n) (2+6 Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,9]+12 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,9])^2)+16 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,9])^3)+25 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,9])^4)+36 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,9])^5)+14 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,9])^6)-40 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,9])^7)-27 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,9])^8)-10 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,9])^9)-22 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,9])^10)+12 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,9])^11)))/((-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,1]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,9]) (-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,2]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,9]) (-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,3]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,9]) (-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,4]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,9]) (-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,5]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,9]) (-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,6]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,9]) (-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,7]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,9]) (-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,8]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,9]) (Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,9]-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,10]) (Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,9]-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,11]) (Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,9]-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,12]))+(((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,10])^n) (2+6 Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,10]+12 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,10])^2)+16 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,10])^3)+25 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,10])^4)+36 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,10])^5)+14 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,10])^6)-40 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,10])^7)-27 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,10])^8)-10 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,10])^9)-22 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,10])^10)+12 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,10])^11)))/((-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,1]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,10]) (-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,2]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,10]) (-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,3]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,10]) (-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,4]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,10]) (-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,5]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,10]) (-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,6]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,10]) (-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,7]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,10]) (-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,8]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,10]) (-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,9]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,10]) (Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,10]-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,11]) (Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,10]-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,12]))+(((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,11])^n) (2+6 Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,11]+12 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,11])^2)+16 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,11])^3)+25 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,11])^4)+36 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,11])^5)+14 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,11])^6)-40 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,11])^7)-27 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,11])^8)-10 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,11])^9)-22 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,11])^10)+12 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,11])^11)))/((-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,1]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,11]) (-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,2]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,11]) (-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,3]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,11]) (-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,4]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,11]) (-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,5]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,11]) (-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,6]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,11]) (-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,7]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,11]) (-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,8]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,11]) (-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,9]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,11]) (-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,10]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,11]) (Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,11]-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,12]))+(((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,12])^n) (2+6 Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,12]+12 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,12])^2)+16 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,12])^3)+25 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,12])^4)+36 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,12])^5)+14 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,12])^6)-40 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,12])^7)-27 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,12])^8)-10 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,12])^9)-22 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,12])^10)+12 ((Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,12])^11)))/((-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,1]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,12]) (-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,2]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,12]) (-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,3]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,12]) (-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,4]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,12]) (-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,5]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,12]) (-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,6]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,12]) (-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,7]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,12]) (-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,8]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,12]) (-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,9]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,12]) (-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,10]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,12]) (-Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,11]+Root[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,12])));

N[terms/.n->0]//Chop

terms[[All,1]]

Table[1+RootSum[1+((#)^2)-((#)^4)+((#)^6)&,#^n&]+RootSum[2+2 #+3 ((#)^2)+4 ((#)^3)+4 ((#)^4)+5 ((#)^5)+6 ((#)^6)+2 ((#)^7)-5 ((#)^8)-3 ((#)^9)-((#)^10)-2 ((#)^11)+((#)^12)&,#^n&],{n,20}]

KirchhoffIndex

seq=Table[GraphData[{"Prism",n},"KirchhoffIndex"],{n,3,25}]

(seq=Table[KirchhoffIndex[PrismGraph[n]],{n,3,50}])//Timing

FindLinearRecurrence[seq]

FindSequenceFunction[seq,n-2]

gf=(x^3)FindGeneratingFunction[seq,x]

KirchhoffSumIndex

seq=Table[GraphData[{"Prism",n},"KirchhoffSumIndex"],{n,3,25}]

(seq=Table[KirchhoffSumIndex[PrismGraph[n]],{n,3,30}])//Timing

FindLinearRecurrence[seq]

FindSequenceFunction[seq,n-2]

gf=(x^3)FindGeneratingFunction[seq,x]

Do[Print[{{"Prism",n},KirchhoffSumIndex[PrismGraph[n]]//Timing}],{n,21,25,2}]

LeafCount

seq=Table[GraphData[{"Prism",n},"LeafCount"],{n,3,25}]

LongestCycleCount

Equal to HamiltonianCyleCount

seq=Table[GraphData[{"Prism",n},"LongestCycleCount"],{n,3,25}]

FindSequenceFunction[seq,n-2]

LovaszNumber

seq=Table[GraphData[{"Prism",n},"LovaszNumber"],{n,3,25}]

LucasCube

Entity["Graph",{"Prism",n}]["LucasCube"]

seq=Table[GraphData[{"Prism",n},"LucasCube"],{n,3,25}]

MatchingNumber

seq=Table[GraphData[{"Prism",n},"MatchingNumber"],{n,3,25}]

FindSequenceFunction[seq,n-2]

MatchingGeneratingPolynomial

Entity["Graph",{"Prism",n}]["MatchingGeneratingPolynomial"]

A102079

(poly=Factor[Table[GraphData[{"Prism",n},"MatchingGeneratingPolynomial"],{n,k0=3,kmax=25}]/.f_Function:>f[x]])//Column

CoefficientList[Take[poly,5],x]

CoefficientList[Table[LinearRecurrence[{1+x,2 x (1+x),-(-1+x) x^2,-x^4},{1+x,1+6 x+5 x^2,1+9 x+18 x^2+4 x^3,(1+4 x+x^2) (1+8 x+9 x^2)},{n}][[1]],{n,3,5}],x]

coef=Factor/@FindLinearRecurrence[DeleteMissing[Take[poly,20]]]

The row generating polynomials A[n] satisfy A[n]=(1+t)A[n-1]+2t(1+t)A[n-2]+ t^2*(1-t)A[n-3]-t^4*A[n-4] with A[2]=1+6t+5t^2, A[3]=1+9t+18t^2+4t^3, A[4]=1+12t+42t^2+44t^3+9t^4 and A[5]=1+15t+75t^2+145t^3+95t^4+11t^5.

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

CoefficientList[Take[poly,5],x]

CoefficientList[Table[(2^(-n))((((1+x-(Sqrt[1+x (6+x)])))^n)+(((1+x+(Sqrt[1+x (6+x)])))^n)),{n,10}]//Expand,x]

SeriesCoefficient[(1+t*z^2)/(1-(1+t)*z-t*z^2),{z,0,n}]

ExpandAll[ ((((1+t-(Sqrt[1+6 t+(t^2)])))^n)+(((1+t+(Sqrt[1+6 t+(t^2)])))^n))/.t->t-1]

ExpandAll[ ((((1+t-(Sqrt[1+6 t+(t^2)])))^n)+(((1+t+(Sqrt[1+6 t+(t^2)])))^n))/.t->t+1]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

gf=(1+t*z^2)/(1-(1+t)*z-t*z^2);

Series[gf,{z,0,10}]

Table[(((-1))^n)(x^n)+(2^(-n))((((1+x-(Sqrt[1+x (6+x)])))^n)+(((1+x+(Sqrt[1+x (6+x)])))^n)),{n,3,5}]//Expand

CoefficientList[Take[poly,5],x]

FullSimplify[SeriesCoefficient[gf,{z,0,n}],n>0]

poly2=Table[(((-1))^n)(x^n)+(2^(-n))((((1+x-(Sqrt[1+x (6+x)])))^n)+(((1+x+(Sqrt[1+x (6+x)])))^n)),{n,k0,kmax}];

FullSimplify[Take[poly2-poly,18]]

MatchingPolynomial

Entity["Graph",{"Prism",n}]["MatchingPolynomial"]

(poly=Factor[Table[GraphData[{"Prism",n},"MatchingPolynomial"],{n,k0=3,kmax=25}]/.f_Function:>f[x]])//Column

CoefficientList[Take[poly,5],x]//Column

coef=Factor/@FindLinearRecurrence[DeleteMissing[Take[poly,20]]]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

Series[gf,{z,0,10}]

CoefficientList[%,z]//Column

FullSimplify[SeriesCoefficient[gf,{z,0,n}],n>2]

MaximalCliqueCount

seq=Table[GraphData[{"Prism",n},"MaximalCliqueCount"],{n,3,25}]

FindSequenceFunction[seq,n-2]

gf=(x^3)FindGeneratingFunction[seq,x]

FullSimplify[SeriesCoefficient[gf,{x,0,n}],n>2]

MaximalIndependentEdgeSetCount

A284703

seq=Table[GraphData[{"Prism",n},"MaximalIndependentEdgeSetCount"],{n,3,25}]

seq={1,5,10,17,51,98,211,457,964,2095,4489,9638,20723,44469,95550,205225,440777,946808,2033571,4367947,9381928,20151345,43283195,92967814,199685501,428904403,921243124,1978737477,4250128177,9128846128,19607840133,42115660425};

FindLinearRecurrence[seq]

seq={1,5,10,17,51,98,211,457,964,2095,4489,9638,20723,44469,95550,205225,440777,946808,2033571,4367947,9381928,20151345,43283195,92967814,199685501,428904403,921243124,1978737477,4250128177,9128846128,19607840133,42115660425};

FindSequenceFunction[DeleteMissing[seq],n]

gf=x FindGeneratingFunction[DeleteMissing[seq],x]

FullSimplify[SeriesCoefficient[gf,{x,0,n}],n>0]

Table[2 Cos[((n π)/3)]+RootSum[-1-2 #-((#)^2)+((#)^3)&,#^n&]+Root[-1-#1+((#1)^3)&,1] ((Root[-1+((#1)^2)+((#1)^3)&,1])^(1+n))+Root[-1-#1+((#1)^3)&,3] ((Root[-1+((#1)^2)+((#1)^3)&,2])^(1+n))+Root[-1-#1+((#1)^3)&,2] ((Root[-1+((#1)^2)+((#1)^3)&,3])^(1+n)),{n,20}]//FullSimplify

FullSimplify[{Root[-1-#1+((#1)^3)&,1] Root[-1+((#1)^2)+((#1)^3)&,1],Root[-1-#1+((#1)^3)&,3] Root[-1+((#1)^2)+((#1)^3)&,2],Root[-1-#1+((#1)^3)&,2] Root[-1+((#1)^2)+((#1)^3)&,3]}]

Table[2 Cos[((n π)/3)]+RootSum[-1-2 #-((#)^2)+((#)^3)&,#^n&]+RootSum[-1+((#)^2)+((#)^3)&,#^n&],{n,20}]

LinearRecurrence[{1,2,1,-1,2,1,-1,-1},{1,5,10,17,51,98,211,457},40]

MaximalIndependentVertexSetCount

A007040

seq=Table[GraphData[{"Prism",n},"MaximalIndependentVertexSetCount"],{n,3,25}]

FindSequenceFunction[Rest@seq,n-3]//FullSimplify

Table[((((1/2) (1-(Sqrt[5]))))^n)+((((1/2) (1+(Sqrt[5]))))^n)+2 Cos[((2 n π)/3)],{n,3,25}]//FullSimplify

FullSimplify[n Sum[Binomial[2k,n-2k]/k,{k,n}]]

gf=(x^3)FindGeneratingFunction[seq,x]

FullSimplify[SeriesCoefficient[gf,{x,0,n}],n>2]

Table[((((1/2) (1-(Sqrt[5]))))^n)+((((1/2) (1+(Sqrt[5]))))^n),{n,10}]//FullSimplify

Table[LucasL[n]+2 Cos[((2 n π)/3)],{n,3,25}]

MaximumCliqueCount

seq=Table[GraphData[{"Prism",n},"MaximumCliqueCount"],{n,3,25}]

FindSequenceFunction[seq,n-2]

gf=(x^3)FindGeneratingFunction[seq,x]

FullSimplify[SeriesCoefficient[gf,{x,0,n}],n>2]

MaximumIndependentEdgeSetCount

A068397

seq=Table[GraphData[{"Prism",n},"MaximumIndependentEdgeSetCount"],{n,3,25}]

FindSequenceFunction[seq,n-2]//FullSimplify

gf=(x^3)FindGeneratingFunction[seq,x]

FullSimplify[SeriesCoefficient[gf,{x,0,n}],n>2]

Table[(-1)^n+LucasL[n]+1,{n,3,25}]

MaximumIndependentVertexSetCount

A266722

seq=Table[GraphData[{"Prism",n},"MaximumIndependentVertexSetCount"],{n,3,25}]

gf=(x^3)FindGeneratingFunction[seq,x]

FullSimplify[SeriesCoefficient[gf,{x,0,n}],n>2]

Table[1-(((-1))^n) (n-1)+n,{n,3,25}]

MaximumLeafNumber

seq=Table[GraphData[{"Prism",n},"MaximumLeafNumber"],{n,3,25}]

MaximumVertexDegree

seq=Table[GraphData[{"Prism",n},"MaximumVertexDegree"],{n,3,25}]

MeanDistance

seq=Table[GraphData[{"Prism",n},"MeanDistance"],{n,3,25}]

FullSimplify[FindSequenceFunction[seq,n-2],n∈Integers]

Table[(((((-1))^n)+2 n (n+2)-1)/(8 n)),{n,3,25}]

gf=(x^3)FindGeneratingFunction[seq,x]//FullSimplify

Series[gf,{x,0,6}]

coef=SeriesCoefficient[gf,{x,0,n}]

Table[coef,{n,3,6}]//FullSimplify

FullSimplify[coef,n>2]

bug(331448): FullSimplify gives wrong result on HypergeometricPFQ + PolyGamma expression

Table[(1/8) (4-(1/n)+2 n),{n,3,25}]

MinimalDominatingSetCount

A290336

seq=Table[GraphData[{"Prism",n},"MinimalDominatingSetCount"],{n,3,25}]

seq=Import["https://oeis.org/A290336/b290336.txt","Table"][[All,2]]

coef=FindLinearRecurrence[DeleteMissing@seq]

Take[seq,Length[coef]]

LinearRecurrence[{1,1,2,-1,2,-2,6,4,4,-6,0,-3,5,-1,-2,-5,-2,-2},{2,4,11,12,37,55,149,316,596,1219,2444,4971,10103,20465,41746,84924,172501,350668},{3,10}]

FindLinearRecurrence[Rest@DeleteMissing@seq]

FindSequenceFunction[DeleteMissing@seq,n]

FindSequenceFunction[DeleteMissing@seq,n-2]

gf=x FindGeneratingFunction[DeleteMissing@seq,x]

Series[gf=x^3*(11+x+14*x^2-16*x^3+44*x^4+28*x^5+56*x^6-52*x^7-6*x^8-70*x^9+52*x^10-28*x^11-23*x^12-97*x^13-56*x^14-62*x^15-12*x^16-8*x^17)/((1-x)*(1+2*x^4+x^6)*(1-x^2-3*x^3-4*x^4-4*x^5-x^6-2*x^7-3*x^8-5*x^9-4*x^10-2*x^11)),{x,0,10}]

res=Refine[SeriesCoefficient[gf,{x,0,n}],n>2]

part1=FullSimplify[RootReduce[2+((E^(-(1/2) I n π)) (-(((-Root[1+2 #1+((#1)^3)&,1]))^(n/2)) (2+3 ((Root[1+2 #1+((#1)^3)&,1])^2)) (Root[1+2 #1+((#1)^3)&,2]-Root[1+2 #1+((#1)^3)&,3])+(E^((I n π)/2)) (-2 ((Root[1+2 #1+((#1)^3)&,1])^(n/2)) (Root[1+2 #1+((#1)^3)&,2]-Root[1+2 #1+((#1)^3)&,3])-3 ((Root[1+2 #1+((#1)^3)&,1])^((4+n)/2)) (Root[1+2 #1+((#1)^3)&,2]-Root[1+2 #1+((#1)^3)&,3])-2 (((-(Sqrt[Root[1+2 #1+((#1)^3)&,2]])))^n) Root[1+2 #1+((#1)^3)&,3]-3 (((-(Sqrt[Root[1+2 #1+((#1)^3)&,2]])))^n) ((Root[1+2 #1+((#1)^3)&,2])^2) Root[1+2 #1+((#1)^3)&,3]-2 ((Root[1+2 #1+((#1)^3)&,2])^(n/2)) Root[1+2 #1+((#1)^3)&,3]-3 ((Root[1+2 #1+((#1)^3)&,2])^((4+n)/2)) Root[1+2 #1+((#1)^3)&,3]+Root[1+2 #1+((#1)^3)&,2] (2+3 ((Root[1+2 #1+((#1)^3)&,3])^2)) ((((-(Sqrt[Root[1+2 #1+((#1)^3)&,3]])))^n)+((Root[1+2 #1+((#1)^3)&,3])^(n/2)))+Root[1+2 #1+((#1)^3)&,1] (2 ((Root[1+2 #1+((#1)^3)&,2])^(n/2))+3 ((Root[1+2 #1+((#1)^3)&,2])^((4+n)/2))+(((-(Sqrt[Root[1+2 #1+((#1)^3)&,2]])))^n) (2+3 ((Root[1+2 #1+((#1)^3)&,2])^2))-(2+3 ((Root[1+2 #1+((#1)^3)&,3])^2)) ((((-(Sqrt[Root[1+2 #1+((#1)^3)&,3]])))^n)+((Root[1+2 #1+((#1)^3)&,3])^(n/2)))))))/(-3+2 Root[1+2 #1+((#1)^3)&,1] ((Root[1+2 #1+((#1)^3)&,2])^2)+2 ((Root[1+2 #1+((#1)^3)&,1])^2) Root[1+2 #1+((#1)^3)&,3]+2 Root[1+2 #1+((#1)^3)&,2] ((Root[1+2 #1+((#1)^3)&,3])^2))],n∈Integers&&n>2]

Expand[FullSimplify[part1/.((r_Root)^(n/2)):>((RootReduce[Sqrt[r]])^n)]/.((r_Root)^(-n)):>((RootReduce[1/r])^n)]

Table[(I^(-n)) (1+(((-1))^n)) ((Root[-1+2 #1+((#1)^3)&,1])^(n/2))+((Root[1+2 #1+((#1)^3)&,2])^(n/2))+((Root[1+2 #1+((#1)^3)&,3])^(n/2))+(I^(-n)) (2 (I^n)+((Root[-1+2 ((#1)^2)+((#1)^6)&,3])^n)+((Root[-1+2 ((#1)^2)+((#1)^6)&,5])^n)),{n,10}]//Round

Table[2+(I^(-n))RootSum[-1+2 ((#1)^2)+((#1)^6)&,#^n&],{n,10}]

Table[2+RootSum[1+2 ((#1)^2)+((#1)^6)&,#^n&],{n,10}]

terms=List@@((((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,1])^n) (-4-10 Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,1]-9 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,1])^2)-8 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,1])^3)-5 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,1])^4)-24 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,1])^5)-28 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,1])^6)-24 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,1])^7)-9 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,1])^8)+11 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,1])^10)))/((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,1]-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,2]) (Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,1]-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,3]) (Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,1]-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,4]) (Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 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((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,9]) (Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,1]-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,10]) (Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,1]-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,11]))+(((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,2])^n) (-4-10 Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,2]-9 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,2])^2)-8 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,2])^3)-5 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,2])^4)-24 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,2])^5)-28 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,2])^6)-24 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,2])^7)-9 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,2])^8)+11 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,2])^10)))/((-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,1]+Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,2]) (Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,2]-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,3]) (Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,2]-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,4]) (Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,2]-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,5]) (Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,2]-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,6]) (Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,2]-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,7]) (Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,2]-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,8]) (Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,2]-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,9]) (Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,2]-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,10]) (Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,2]-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,11]))+(((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,3])^n) (-4-10 Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,3]-9 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,3])^2)-8 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,3])^3)-5 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,3])^4)-24 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,3])^5)-28 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,3])^6)-24 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,3])^7)-9 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,3])^8)+11 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,3])^10)))/((-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,1]+Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,3]) (-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,2]+Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,3]) (Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,3]-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,4]) (Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,3]-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,5]) (Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,3]-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,6]) (Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,3]-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,7]) (Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,3]-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,8]) (Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,3]-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,9]) (Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,3]-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,10]) (Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,3]-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,11]))+(((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,4])^n) (-4-10 Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,4]-9 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,4])^2)-8 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,4])^3)-5 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,4])^4)-24 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,4])^5)-28 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,4])^6)-24 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,4])^7)-9 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,4])^8)+11 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,4])^10)))/((-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,1]+Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,4]) (-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,2]+Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,4]) (-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,3]+Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,4]) (Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,4]-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,5]) (Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,4]-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,6]) (Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,4]-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,7]) (Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,4]-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,8]) (Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,4]-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,9]) (Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,4]-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,10]) (Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,4]-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,11]))+(((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,5])^n) (-4-10 Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,5]-9 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,5])^2)-8 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,5])^3)-5 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,5])^4)-24 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,5])^5)-28 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,5])^6)-24 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,5])^7)-9 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,5])^8)+11 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,5])^10)))/((-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,1]+Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,5]) (-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,2]+Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,5]) (-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,3]+Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,5]) (-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,4]+Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,5]) (Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,5]-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,6]) (Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,5]-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,7]) (Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,5]-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,8]) (Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,5]-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,9]) (Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,5]-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,10]) (Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,5]-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,11]))+(((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,6])^n) (-4-10 Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,6]-9 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,6])^2)-8 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,6])^3)-5 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,6])^4)-24 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,6])^5)-28 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,6])^6)-24 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,6])^7)-9 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,6])^8)+11 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,6])^10)))/((-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,1]+Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,6]) (-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,2]+Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,6]) (-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,3]+Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,6]) (-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,4]+Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,6]) (-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,5]+Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,6]) (Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,6]-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,7]) (Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,6]-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,8]) (Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,6]-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,9]) (Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,6]-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,10]) (Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,6]-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,11]))+(((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,7])^n) (-4-10 Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,7]-9 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,7])^2)-8 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,7])^3)-5 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,7])^4)-24 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,7])^5)-28 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,7])^6)-24 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,7])^7)-9 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,7])^8)+11 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,7])^10)))/((-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,1]+Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,7]) (-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,2]+Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,7]) (-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,3]+Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,7]) (-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,4]+Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,7]) (-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,5]+Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,7]) (-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,6]+Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,7]) (Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,7]-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,8]) (Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,7]-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,9]) (Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,7]-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,10]) (Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,7]-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,11]))+(((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,8])^n) (-4-10 Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,8]-9 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,8])^2)-8 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,8])^3)-5 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,8])^4)-24 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,8])^5)-28 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,8])^6)-24 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,8])^7)-9 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,8])^8)+11 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,8])^10)))/((-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,1]+Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,8]) (-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,2]+Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,8]) (-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,3]+Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,8]) (-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,4]+Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,8]) (-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,5]+Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,8]) (-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,6]+Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,8]) (-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,7]+Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,8]) (Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,8]-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,9]) (Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,8]-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,10]) (Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,8]-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,11]))+(((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,9])^n) (-4-10 Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,9]-9 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,9])^2)-8 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,9])^3)-5 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,9])^4)-24 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,9])^5)-28 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,9])^6)-24 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,9])^7)-9 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,9])^8)+11 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,9])^10)))/((-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,1]+Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,9]) (-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,2]+Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,9]) (-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,3]+Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,9]) (-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,4]+Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,9]) (-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,5]+Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,9]) (-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,6]+Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,9]) (-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,7]+Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,9]) (-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,8]+Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,9]) (Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,9]-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,10]) (Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,9]-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,11]))+(((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,10])^n) (-4-10 Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,10]-9 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,10])^2)-8 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,10])^3)-5 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,10])^4)-24 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,10])^5)-28 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,10])^6)-24 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,10])^7)-9 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,10])^8)+11 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,10])^10)))/((-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,1]+Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,10]) (-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,2]+Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,10]) (-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,3]+Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,10]) (-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,4]+Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,10]) (-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,5]+Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,10]) (-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,6]+Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,10]) (-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,7]+Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,10]) (-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,8]+Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,10]) (-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,9]+Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,10]) (Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,10]-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,11]))+(((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,11])^n) (-4-10 Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,11]-9 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,11])^2)-8 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,11])^3)-5 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,11])^4)-24 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,11])^5)-28 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,11])^6)-24 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,11])^7)-9 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,11])^8)+11 ((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,11])^10)))/((-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,1]+Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,11]) (-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,2]+Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,11]) (-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,3]+Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,11]) (-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,4]+Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,11]) (-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,5]+Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,11]) (-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,6]+Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,11]) (-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,7]+Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,11]) (-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,8]+Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,11]) (-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,9]+Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,11]) (-Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,10]+Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,11])));

N[terms/.n->0]//Chop

terms[[All,1]]

Table[2+RootSum[1+2 ((#)^2)+((#)^6)&,#^n&]+RootSum[-2-4 #-5 ((#)^2)-3 ((#)^3)-2 ((#)^4)-((#)^5)-4 ((#)^6)-4 ((#)^7)-3 ((#)^8)-((#)^9)+((#)^11)&,#^n&],{n,10}]

MinimalDominatingSetSignature

seq=Table[GraphData[{"Prism",n},"MinimalDominatingSetSignature"],{n,3,25}]//Column

FindSequenceFunction[DeleteMissing@seq,n-2]//Factor

MinimalDominationPolynomial

Entity["Graph",{"Prism",n}]["MinimalDominationPolynomial"]

(poly=Factor[Table[GraphData[{"Prism",n},"MinimalDominationPolynomial"],{n,k0=3,kmax=25}]/.f_Function:>f[x]])//Column

CoefficientList[Take[poly,5],x]//Column

coef=Factor/@FindLinearRecurrence[DeleteMissing[Take[poly,20]]]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

MinimalEdgeCoverCount

Entity["Graph",{"Prism",n}]["MinimalEdgeCoverCount"]

A290700

seq=Table[GraphData[{"Prism",n},"MinimalEdgeCoverCount"],{n,3,25}]

Table[GraphData[{"Prism",n},"EdgeCoverCount"],{n,3,25}]

seq={1,5,25,49,141,389,1009,2761,7441,19925,53769,144721,389325,1048325,2821665,7594761,20444065,55029413,148124153,398713969,1073231821,2888859781,7776063377,20931130057,56341150641,151655712629,408217654249,1098815597201};

coef=FindLinearRecurrence[DeleteMissing@seq]

Take[seq,Length@coef]

LinearRecurrence[{1,2,6,2,2,-2,-2,-1,1},{1,5,25,49,141,389,1009,2761,7441},20]

FindSequenceFunction[DeleteMissing@seq,n]

gf=x FindGeneratingFunction[DeleteMissing@seq,x]//Factor

CoefficientList[Series[-(( (1+4 x+18 (x^2)+8 (x^3)+10 (x^4)-12 (x^5)-14 (x^6)-8 (x^7)+9 (x^8)))/((1+(x^2)) (-1-x-(x^2)+(x^3)) (1-2 x-2 (x^2)+(x^4)))),{x,0,20}],x]

res=Refine[SeriesCoefficient[gf,{x,0,n}],n>0]

FullSimplify[RootReduce[res],n∈Integers&&n>0]

terms=List@@(Root[1-2 #1-2 ((#1)^2)+((#1)^4)&,2] ((Root[1-2 ((#1)^2)-2 ((#1)^3)+((#1)^4)&,1])^(1+n))+Root[1-2 #1-2 ((#1)^2)+((#1)^4)&,1] ((Root[1-2 ((#1)^2)-2 ((#1)^3)+((#1)^4)&,2])^(1+n))+Root[1-2 #1-2 ((#1)^2)+((#1)^4)&,4] ((Root[1-2 ((#1)^2)-2 ((#1)^3)+((#1)^4)&,3])^(1+n))+Root[1-2 #1-2 ((#1)^2)+((#1)^4)&,3] ((Root[1-2 ((#1)^2)-2 ((#1)^3)+((#1)^4)&,4])^(1+n)));

pos=(terms/.n->-1)[[All,2]]

f1=1-2 ((#1)^2)-2 ((#1)^3)+((#1)^4)&;
f2=1-2 #1-2 ((#1)^2)+((#1)^4)&;

Table[2 Cos[((n π)/2)]+RootSum[-1+#1+((#1)^2)+((#1)^3)&,#^n&]+Sum[((Root[f1,k])^(n+1))Root[f2,pos[[k]]],{k,Length[pos]}],{n,10}]//N//Chop

an=ToNumberField[Root[f2,pos[[#]]],Root[f1,#]]&/@Range[Length[pos]];

Equal@@Last/@an

RootSum[f1,Evaluate[((#)^(n+Range[Length[pos]])).an[[1,2]]]&]

Table[2 Cos[((n π)/2)]+RootSum[-1+#+((#)^2)+((#)^3)&,#^n&]-RootSum[1-2 ((#)^2)-2 ((#)^3)+((#)^4)&,-2 ((#)^(n+2))-2 ((#)^(n+3))+((#)^(n+4))&],{n,20}]

MinimalTotalDominatingSetCount

Entity["Graph",{"Prism",n}]["MinimalTotalDominatingSetCount"]

seq=Table[GraphData[{"Prism",n},"MinimalTotalDominatingSetCount"],{n,3,25}]

Table[3+3 (((-1))^n)+RootSum[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,((#1)^n)&]+RootSum[1-4 #1+10 ((#1)^2)-19 ((#1)^3)+28 ((#1)^4)-34 ((#1)^5)+37 ((#1)^6)-34 ((#1)^7)+28 ((#1)^8)-19 ((#1)^9)+10 ((#1)^10)-4 ((#1)^11)+((#1)^12)&,((#1)^n)&]+RootSum[1+4 #1+10 ((#1)^2)+19 ((#1)^3)+28 ((#1)^4)+34 ((#1)^5)+37 ((#1)^6)+34 ((#1)^7)+28 ((#1)^8)+19 ((#1)^9)+10 ((#1)^10)+4 ((#1)^11)+((#1)^12)&,((#1)^n)&],{n,3,25}]

(poly=Factor[Table[GraphData[{"Prism",n},"MatchingGeneratingPolynomial"],{n,k0=3,kmax=25}]/.f_Function:>f[x]])//Column

MinimalTotalDominationPolynomial

Entity["Graph",{"Prism",n}]["MinimalTotalDominationPolynomial"]

(poly=Factor[Table[GraphData[{"Prism",n},"MinimalTotalDominationPolynomial"],{n,k0=3,kmax=25}]/.f_Function:>f[x]])//Column

CoefficientList[Take[poly,5],x]//Column

coef=Factor/@FindLinearRecurrence[DeleteMissing[Take[poly,20]]]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

MinimumCliqueCoveringCount

A068397 (except for first term)

seq=Table[GraphData[{"Prism",n},"MinimumCliqueCoveringCount"],{n,3,25}]

FindSequenceFunction[DeleteMissing[seq],n-2]

Table[LucasL[n]+(-1)^n+1,{n,3,25}]

Table[Piecewise[{{1,n==3}},LucasL[n]+(-1)^n+1],{n,3,25}]

LucasL[#]+(-1)^#+1&[Range[38]]

MinimumConnectedDominatingSetCount

seq=Table[GraphData[{"Prism",n},"MinimumConnectedDominatingSetCount"],{n,3,25}]

FindSequenceFunction[DeleteMissing[seq],n-2]

gf=(x^3)FindGeneratingFunction[DeleteMissing[seq],x]

FullSimplify[SeriesCoefficient[gf,{x,0,n}],n>2]

Table[Piecewise[{{3,n==3},{30,n==4}},2(n^2+1)],{n,3,25}]

MinimumDistinguishingLabelingCount

seq=Table[GraphData[{"Prism",n},"MinimumDistinguishingLabelingCount"],{n,3,25}]

FindLinearRecurrence[DeleteMissing[seq]]

FindSequenceFunction[DeleteMissing[seq],n-2]

gf=(x^3) FindGeneratingFunction[DeleteMissing[seq],x]

MinimumDominatingSetCount

seq=Table[GraphData[{"Prism",n},"MinimumDominatingSetCount"],{n,3,25}]

seq[[2;;All;;4]]

seq[[3;;All;;2]]

seq[[4;;All;;4]]

Table[8n(2n-1),{n,10}]

Table[Piecewise[{{9,n==3},{51,n==6},{4,Mod[n,4]==0},{2n,Mod[n,2]==1},{n(n+2),Mod[n,4]==2}}],{n,3,25}]

Tally[Length/@FindDominatingSet[GraphData[{"Prism",6}],Infinity,All]]

GraphData[{"Prism",6},"DominationPolynomial"][x]

GraphData[{"Prism",6},"Graph","Labeled"]

GraphData[{"Prism",6},"MinimumDominatingSets"]

GraphData[{"Prism",6},"MinimumDominatingSetCount"]

GraphData[{"Prism",7},"MinimumDominatingSetCount"]

GraphData[{"Prism",8},"Graph","Labeled"]

GraphData[{"Prism",8},"MinimumDominatingSets"]

MinimumEdgeCoverCount

Same as MaximumIndependentEdgeSetCount

seq=Table[GraphData[{"Prism",n},"MinimumEdgeCoverCount"],{n,3,25}]

FindSequenceFunction[seq,n-2]//FullSimplify

Table[(-1)^n+LucasL[n]+1,{n,3,25}]

MinimumTotalDominatingSetCount

Entity["Graph",{"Prism",n}]["MinimumTotalDominatingSetCount"]

seq=Table[GraphData[{"Prism",n},"MinimumTotalDominatingSetCount"],{n,3,25}]

MinimumVertexCoverCount

Entity["Graph",{"Prism",n}]["MinimumVertexCoverCount"]

Same as MaximumIndependentVertexSetCount

seq=Table[GraphData[{"Prism",n},"MinimumVertexCoverCount"],{n,3,25}]

gf=(x^3)FindGeneratingFunction[seq,x]

FullSimplify[SeriesCoefficient[gf,{x,0,n}],n>2]

MinimumVertexDegree

seq=Table[GraphData[{"Prism",n},"MinimumVertexDegree"],{n,3,25}]

MolecularTopologicalIndex

A192838

seq=Table[GraphData[{"Prism",n},"MolecularTopologicalIndex"],{n,3,25}]

FindLinearRecurrence[seq]

LinearRecurrence[{2,1,-4,1,2,-1},{24,84,180,360,600,972,1428},20]

FindSequenceFunction[seq,n-2]//FullSimplify

Table[3/2 n (11+(-1)^n+2 n (n+2)),{n,20}]

Nonhamiltonian

Table[GraphData[{"Prism",n},"Nonhamiltonian"],{n,3,25}]

NotationRules

Table[GraphData[{"Prism",n},"NotationRules","Prism"],{n,3,25}]

Table[GraphData[{"Prism",n},"NotationRules","Polyhedral"],{n,3,25}]

seq=DeleteMissing[%][[All,-1]]

FindGeneratingFunction[seq,x]//Head//Timing

OddChordlessCycleCount

seq=Table[GraphData[{"Prism",n},"OddChordlessCycleCount"],{n,3,25}]

seq[[3;;All;;2]]

Total/@{
{{17,2},{19,238},{21,1870},{23,1428},{25,34}},
{{19,2},{21,304},{23,3458},{25,5016},{27,570}},
{{21,2},{23,378},{25,5880},{27,14014},{29,4158},{31,42}},
{{23,2},{25,460},{27,9384},{29,33488},{31,19734},{33,1012}}
}[[All,All,2]]

FindSequenceFunction[seq={12,30,74,200,522,1362,3572,9350,24474,64080},n]//FullSimplify

Table[(2^(-n)) (-(((3-(Sqrt[5])))^n) (-2+(Sqrt[5]))+(2+(Sqrt[5])) (((3+(Sqrt[5])))^n))-2 Cos[((2 n π)/3)],{n,4}]//FullSimplify

Table[((2^(-n)) (-(((3-(Sqrt[5])))^n) (-2+(Sqrt[5]))+(2+(Sqrt[5])) (((3+(Sqrt[5])))^n))-2 Cos[((2 n π)/3)])/.n->(k-3)/2,{k,5,10,2}]//FullSimplify

FullSimplify[((2^(-n)) (-(((3-(Sqrt[5])))^n) (-2+(Sqrt[5]))+(2+(Sqrt[5])) (((3+(Sqrt[5])))^n))-2 Cos[((2 n π)/3)])/.n->(n-3)/2]

{((1/2) (3-(Sqrt[5]))),((1/2) (3+(Sqrt[5])))}//N

{1/GoldenRatio,GoldenRatio}^2//N

Table[Piecewise[{{0,n==3},{0,Mod[n,2]==0},{LucasL[n]+2 Cos[((n π)/3)],Mod[n,2]==1}}],{n,3,23}]//FullSimplify

Table[(GoldenRatio^n)-(GoldenRatio^(-n)),{n,1,10,2}]//FullSimplify

Table[LucasL[n],{n,1,10,2}]//FullSimplify

gf=(x^3)FindGeneratingFunction[seq,x]

FullSimplify[SeriesCoefficient[gf,{x,0,n}],n>2]

%/2

PathCount

A287992

seq=Table[GraphData[{"Prism",n},"PathCount"],{n,3,20}]

FindLinearRecurrence[DeleteMissing@seq]

FindSequenceFunction[DeleteMissing@seq,n-2]

gf=(x^3) FindGeneratingFunction[DeleteMissing@seq,x]

FullSimplify[SeriesCoefficient[gf,{x,0,n}],n>2]//Factor

Table[(5 (2^(n+1))-5 n-(n^2)-13)n,{n,30}]

PathPolynomial

(poly=Factor[Table[GraphData[{"Prism",n},"PathPolynomial"],{n,k0=3,kmax=25}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[DeleteMissing@poly]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

FullSimplify[SeriesCoefficient[gf,{z,0,n}],n>0]

Take[poly2,1]//Factor

poly2=Table[-(1/((-1+2 x) (((-1+(x^2)))^2) (-1+x+(x^2))))n x (-2 (-3+n) (x^(2+2 n))+2 (5+3 n) (x^(3+2 n))-10 (x^(4+2 n))-2 (4+5 n) (x^(5+2 n))+2 (-2+n) (x^(6+2 n))+4 (1+n) (x^(7+2 n))+4 (x^(8+2 n))+(((-(-1+x) x))^n)+(((x (1+x)))^n)-(x^5) (9+(((-(-1+x) x))^n)-15 (((x (1+x)))^n))+(x^3) (9+2 (((-(-1+x) x))^n)-12 (((x (1+x)))^n))+(x^7) (1-4 (((x (1+x)))^n))+(x^8) (2-4 (((x (1+x)))^n))-x (3+4 (x^(2 n))+(((-(-1+x) x))^n)-(((x (1+x)))^n))+(x^6) (1-(((-(-1+x) x))^n)+(((x (1+x)))^n))-3 (x^2) (-1+(((-(-1+x) x))^n)+(((x (1+x)))^n))+(x^4) (-2+3 (((-(-1+x) x))^n)+5 (((x (1+x)))^n))),{n,k0,kmax}]

Factor[Take[poly2-poly,kmax-k0-Count[poly,_Missing]+1]]

Pell

Entity["Graph",{"Prism",n}]["Pell"]

seq=Table[GraphData[{"Prism",n},"Pell"],{n,3,25}]

PentagonCount

seq=Table[GraphData[{"Prism",n},"PentagonCount"],{n,3,25}]

Perfect

seq=Table[GraphData[{"Prism",n},"Perfect"],{n,3,25}]

ProjectivePlaneCrossingNumber

seq=Table[GraphData[{"Prism",n},"ProjectivePlaneCrossingNumber"],{n,3,25}]

Radius

seq=Table[GraphData[{"Prism",n},"Radius"],{n,3,25}]

Table[1+Floor[n/2],{n,3,25}]

RectilinearCrossingNumber

seq=Table[GraphData[{"Prism",n},"RectilinearCrossingNumber"],{n,3,25}]

Skewness

seq=Table[GraphData[{"Prism",n},"Skewness"],{n,3,25}]

SpanningTreeCount

A006235

seq=Table[GraphData[{"Prism",n},"SpanningTreeCount"],{n,3,25}]

FindSequenceFunction[seq,n-2]//FullSimplify

gf=(x^3)FindGeneratingFunction[seq,x]

FullSimplify[SeriesCoefficient[gf,{x,0,n}],n>2]

(1/2)Table[(((2-(Sqrt[3])))^n)+(((2+(Sqrt[3])))^n),{n,10}]//FullSimplify

Table[ChebyshevT[n,2],{n,10}]

Table[n(ChebyshevT[n,2]-1),{n,20}]

SquareCount

seq=Table[GraphData[{"Prism",n},"SquareCount"],{n,3,25}]

StabilityIndex

seq=Table[GraphData[{"Prism",n},"StabilityIndex"],{n,3,25}]

(seq=Table[StabilityIndex[PrismGraph[n]],{n,3,40}])//Timing

coef=FindLinearRecurrence[seq]

Length[coef]

LinearRecurrence[coef,seq,{-1,Length[coef]-2}]

Length[%]

eqns=Join[{a[n]==coef.Table[a[n-k],{k,Length[coef]}]},MapIndexed[a[#2[[1]]+2]==#&,Take[seq,Length[coef]]]]

RSolve[eqns,a[n],n]//Timing

FindSequenceFunction[seq,n-2]

gf=(x^3)FindGeneratingFunction[seq,x]

TopologicalIndex

seq=Table[GraphData[{"Prism",n},"TopologicalIndex"],{n,3,25}]

FindSequenceFunction[seq,n-2]

(x^3)FindGeneratingFunction[seq,x]

(seq=Table[TopologicalIndex[PrismGraph[n]],{n,3,kmax=70}])//Timing

FindLinearRecurrence[seq]

FindLinearRecurrence[Abs[seq]]

Table[{n,seq[[n-2]]},{n,3,kmax,6}]

FindSequenceFunction[%,n]

Table[{n,seq[[n-2]]},{n,4,kmax,6}]

FindSequenceFunction[%,n]

Table[{n,seq[[n-2]]},{n,5,kmax,6}]

FindSequenceFunction[%,n]

Table[{n,seq[[n-2]]},{n,6,kmax,6}]

FindSequenceFunction[%,n]

Table[{n,seq[[n-2]]},{n,7,kmax,6}]

FindSequenceFunction[%,n]

ToroidalCrossingNumber

seq=Table[GraphData[{"Prism",n},"ToroidalCrossingNumber"],{n,3,25}]

TotalDominatingSetCount

Entity["Graph",{"Prism",n}]["TotalDominatingSetCount"]

seq=Table[GraphData[{"Prism",n},"TotalDominatingSetCount"],{n,3,25}]

TotalDominationNumber

Entity["Graph",{"Prism",n}]["TotalDominationNumber"]

seq=Table[GraphData[{"Prism",n},"TotalDominationNumber"],{n,3,25}]

TotalDominationPolynomial

Entity["Graph",{"Prism",n}]["TotalDominationPolynomial"]

(poly=Factor[Table[GraphData[{"Prism",n},"TotalDominationPolynomial"],{n,k0=3,kmax=25}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[DeleteMissing[Take[poly,20]]]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

Traceable

Table[GraphData[{"Prism",n},"Traceable"],{n,3,25}]

TriangleCount

seq=Table[GraphData[{"Prism",n},"TriangleCount"],{n,3,25}]

TuttePolynomial

Entity["Graph",{"Prism",n}]["TuttePolynomial"]

(poly=Factor@Table[GraphData[{"Prism",n},"TuttePolynomial"],{n,k0=3,kmax=25}]/.f_Function:>f[x,y])//Column

coef=Factor/@FindLinearRecurrence[DeleteMissing[poly]]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

FullSimplify[SeriesCoefficient[gf,{z,0,n}],n>2]

Untraceable

Table[GraphData[{"Prism",n},"Untraceable"],{n,3,25}]

VertexConnectivity

Table[GraphData[{"Prism",n},"VertexConnectivity"],{n,3,25}]

VertexCoverCount

Same as IndependentVertexSetCount

seq=Table[GraphData[{"Prism",n},"VertexCoverCount"],{n,3,25}]

FindSequenceFunction[seq,n-2]//FullSimplify

Table[(-1)^n+LucasL[n,2],{n,3,25}]

VertexCoverNumber

Related to but not the same as IndependenceNumber

seq=Table[GraphData[{"Prism",n},"VertexCoverNumber"],{n,3,25}]

FindSequenceFunction[seq,n-2]//FullSimplify

Table[2Floor[(n+1)/2],{n,3,25}]

gf=x FindGeneratingFunction[seq,x]

FullSimplify[SeriesCoefficient[gf,{x,0,n}],n>0]

Table[3 (2^(-3+n)),{n,3,25}]

VertexCount

seq=Table[GraphData[{"Prism",n},"VertexCount"],{n,3,25}]

FindSequenceFunction[seq,n-2]

VertexDegrees

seq=Table[GraphData[{"Prism",n},"VertexDegrees"],{n,3,5}]

WienerIndex

A138179

seq=Table[GraphData[{"Prism",n},"WienerIndex"],{n,3,25}]

coef=FindLinearRecurrence[seq]

Take[seq,Length@coef]

LinearRecurrence[{2,1,-4,1,2,-1},{21,48,85,144,217,320},{-1,1}]

seq=LinearRecurrence[{2,1,-4,1,2,-1},{1,8,21,48,85,144},20]

FindSequenceFunction[seq,n]//FullSimplify

Table[(1/4) (((-1))^n) n (1+(((-1))^n) (-1+2 n (2+n))),{n,25}]

gf=x FindGeneratingFunction[seq,x]

CoefficientList[Series[((1+6 x+4 (x^2)+2 (x^3)-(x^4))/((((-1+x))^4) (((1+x))^2))),{x,0,20}],x]

WienerSumIndex

seq=Table[GraphData[{"Prism",n},"WienerSumIndex"],{n,3,25}]

FindSequenceFunction[seq,n-2]

gf=(x^3)FindGeneratingFunction[seq,x]

GraphData

GraphDataString

file=OpenWrite["~/prism.m"]

Function[n,WriteString[file,GraphDataString[PrismGraph[n],
"StandardName"->{"Prism",n},
"AlternateStandardNames"->{{"CircularLadder",n},{"GeneralizedPrism",{n,2}},{"Prism",{n,2}},{"StackedPrism",{n,2}},"Y"<>ToString[n]},
"Name"->ToString[n]<>"\[Hyphen]prism graph",
"Information"->"PrismGraph",
"Classes"->{"Graceful"},
"GracefulLabeling"->GracefulLabeling[{"Prism",n}],
"AlternateNames"->{"("<>ToString[n]<>",2)\[Hyphen]generalized prism graph",ToString[n]<>"\[Hyphen]circular ladder graph","icosi"<>Switch[n,20,"",21,"hena",22,"di",23,"tri",24,"tetra",25,"penta"]<>"gonal prism graph"},
"Notation"->Row[{Subscript["P",2],"\[Square]",Subscript["C",n]}],
"NotationRules"->{"Cp"->Missing["NotAvailable"],"Ct"->None,"Prism"->n,"StackedPrism"->{n,2}},
"RegularParameters"->{2n,{3},{0},{0,1,2}},
TimeConstraint->60
]]]/@{22,24};

Close[file]