Pan Graph

Author

Eric W. Weisstein
March 3, 2018

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

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

©2018 Wolfram Research, Inc. except for portions noted otherwise

Figures

"BannerGraph"

GraphData["BannerGraph","Graph","All"]//StyleGraphs

Graphs.m

Default

GraphicsRow[Table[Show[PanGraph[n]//StyleGraphs[#,VertexSize->.25]&,PlotLabel->Style[HoldForm[n]==n,FontFamily->"Times"]],{n,3,7}],Spacings->-30,Dividers->All]

"UnitDistance"

GraphicsRow[Table[Show[PanGraph[n,GraphLayout->"UnitDistance"]//StyleGraphs[#,VertexSize->.25]&,PlotLabel->Style[HoldForm[n]==n,FontFamily->"Times"]],{n,3,7}],Spacings->-30,Dividers->All]

GraphData

Primary

GraphicsGrid[Partition[Table[Show[GraphData[{"Pan",n}]//StyleGraphs,PlotLabel->Style[HoldForm[n]==n,FontFamily->"Times"]],{n,3,20}],5],Dividers->All,ImageSize->600]

"UnitDistance"

GraphicsGrid[Partition[Table[Show[GraphData[{"Pan",n},"Graph","UnitDistance"][[1]]//StyleGraphs,PlotLabel->Style[HoldForm[n]==n,FontFamily->"Times"]],{n,3,20}],5],Dividers->All,ImageSize->600]

Embeddings

4-pan graph

GraphData[{"Pan",4},"#"]

GraphData[{"Pan",3},"Graph","All"]

GraphData[{"Pan",4},"Graph","All"]

GraphData[{"Pan",4},"EmbeddingClasses"]

GraphData

Primary

GraphicsGrid[Partition[Table[GraphData[{"Pan",n}],{n,3,20}],UpTo[5]],ImageSize->600]//StyleGraphs

UnitDistance

GraphicsGrid[Partition[Table[GraphData[{"Pan",n},"Graph","UnitDistance"][[1]],{n,3,20}],UpTo[5]],ImageSize->600]//StyleGraphs

Graphs.m

Automatic

GraphicsGrid[Partition[Table[PanGraph[n],{n,3,20}],UpTo[5]],ImageSize->600]//StyleGraphs

UnitDistance

GraphicsGrid[Partition[Table[PanGraph[n,GraphLayout->"UnitDistance"],{n,3,20}],UpTo[5]],ImageSize->600]//StyleGraphs

Properties

Acyclic

seq=Table[GraphData[{"Pan",n},"Acyclic"],{n,3,20}]

AdjacencyMatrixCount

seq=Table[GraphData[{"Pan",n},"AdjacencyMatrixCount"],{n,3,20}]

Arboricity

seq=Table[GraphData[{"Pan",n},"Arboricity"],{n,3,20}]

ArcTransitive

seq=Table[GraphData[{"Pan",n},"ArcTransitive"],{n,3,20}]

ArcTransitivity

seq=Table[GraphData[{"Pan",n},"ArcTransitivity"],{n,3,20}]

AutomorphismCount

seq=Table[GraphData[{"Pan",n},"AutomorphismCount"],{n,3,20}]

BalabanIndex

seq=Table[GraphData[{"Pan",n},"BalabanIndex"],{n,3,6}]//FullSimplify

Bandwidth

seq=Table[GraphData[{"Pan",n},"Bandwidth"],{n,3,20}]

BridgeCount

seq=Table[GraphData[{"Pan",n},"BridgeCount"],{n,3,20}]

BurningNumber

seq=Table[GraphData[{"Pan",n},"BurningNumber"],{n,3,20}]

CharacteristicPolynomial

(poly=Factor[Table[GraphData[{"Pan",n},"CharacteristicPolynomial"],{n,k0=3,kmax=20}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[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,k0,kmax}]

FullSimplify[poly2-poly]

Chordal

seq=Table[GraphData[{"Pan",n},"Chordal"],{n,3,20}]

ChordCount

seq=Table[GraphData[{"Pan",n},"ChordCount"],{n,3,20}]

Chordless

seq=Table[GraphData[{"Pan",n},"Chordless"],{n,3,20}]

ChordlessCycleCount

seq=Table[GraphData[{"Pan",n},"ChordlessCycleCount"],{n,3,20}]

Chords

seq=Table[GraphData[{"Pan",n},"Chords"],{n,3,20}]

ChromaticInvariant

seq=Table[GraphData[{"Pan",n},"ChromaticInvariant"],{n,3,20}]

ChromaticNumber

seq=Table[GraphData[{"Pan",n},"ChromaticNumber"],{n,3,20}]

ChromaticPolynomial

(poly=Factor[Table[GraphData[{"Pan",n},"ChromaticPolynomial"],{n,k0=3,kmax=20}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[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,k0,kmax}]

FullSimplify[poly2-poly]

Circumference

seq=Table[GraphData[{"Pan",n},"Circumference"],{n,3,20}]

FindSequenceFunction[seq,n-2]

Classes

Intersection@@(GraphData[#,"Classes"]&/@GraphData["Pan"])

CliqueCount

seq=Table[GraphData[{"Pan",n},"CliqueCount"],{n,3,20}]

FindSequenceFunction[seq,n-2]

gf=x FindGeneratingFunction[seq,x]

SeriesCoefficient[gf,{x,0,n}]//FullSimplify

CliqueCoveringNumber

seq=Table[GraphData[{"Pan",n},"CliqueCoveringNumber"],{n,3,20}]

FindSequenceFunction[Rest@seq,n-3]//FullSimplify

gf=x FindGeneratingFunction[seq,x]

FullSimplify[SeriesCoefficient[gf,{x,0,n}],n>0]

CliqueNumber

seq=Table[GraphData[{"Pan",n},"CliqueNumber"],{n,3,20}]

CliquePolynomial

(poly=Factor[Table[GraphData[{"Pan",n},"CliquePolynomial"],{n,k0=4,kmax=20}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[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,k0,kmax}]

FullSimplify[poly2-poly]

ComplementOddChordlessCycleCount

seq=Table[GraphData[{"Pan",n},"ComplementOddChordlessCycleCount"],{n,3,20}]

ConnectedComponentCount

seq=Table[GraphData[{"Pan",n},"ConnectedComponentCount"],{n,3,20}]

ConnectedDominatingSetCount

seq=Table[GraphData[{"Pan",n},"ConnectedDominatingSetCount"],{n,3,20}]

FindSequenceFunction[seq,n-2]

ConnectedDominationNumber

seq=Table[GraphData[{"Pan",n},"ConnectedDominationNumber"],{n,3,20}]

FindSequenceFunction[seq,n-2]

ConnectedDominationPolynomial

(poly=Factor[Table[GraphData[{"Pan",n},"ConnectedDominationPolynomial"],{n,k0=3,kmax=20}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[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,k0,kmax}]

FullSimplify[poly2-poly]

ConnectedInducedSubgraphCount

A290768

seq=Table[GraphData[{"Pan",n},"ConnectedInducedSubgraphCount"],{n,3,20}]

FindLinearRecurrence[DeleteMissing@seq]

seq=LinearRecurrence[{3,-3,1},{12,21,33},{-1,20}]

FindSequenceFunction[DeleteMissing@seq,n]//Factor

Table[3/2(n^2-n+2),{n,50}]

gf=x FindGeneratingFunction[DeleteMissing@seq,x]

CoefficientList[Series[-((3 (1-x+(x^2)))/(((-1+x))^3)),{x,0,20}],x]

ConnectedInducedSubgraphPolynomial

(poly=Factor[Table[GraphData[{"Pan",n},"CharacteristicPolynomial"],{n,k0=3,kmax=20}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[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,6}]

FullSimplify[poly2-poly]

CrossingNumber

seq=Table[GraphData[{"Pan",n},"CrossingNumber"],{n,3,20}]

CycleCount

seq=Table[GraphData[{"Pan",n},"CycleCount"],{n,3,20}]

CyclePolynomial

(poly=Factor[Table[GraphData[{"Pan",n},"CyclePolynomial"],{n,k0=3,kmax=20}]/.f_Function:>f[x]])

Cyclic

Table[GraphData[{"Pan",n},"Cyclic"],{n,3,20}]

CyclomaticNumber

seq=Table[GraphData[{"Pan",n},"CyclomaticNumber"],{n,3,20}]

DetourIndex

seq=Table[GraphData[{"Pan",n},"DetourIndex"],{n,3,20}]

FindSequenceFunction[seq,n-2]//FullSimplify

DetourMatrix

MatrixForm/@Table[GraphData[{"Pan",n},"DetourMatrix"],{n,3,20}]

MatrixForm/@Table[SparseArray[{i_,j_}:>Abs[i-j],n],{n,3,20}]

DetourPolynomial

(poly=Factor[Table[GraphData[{"Pan",n},"DetourPolynomial"],{n,k0=3,kmax=20}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[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,k0,kmax}]

FullSimplify[poly2-poly]

Diameter

seq=Table[GraphData[{"Pan",n},"Diameter"],{n,3,20}]

DistanceMatrix

MatrixForm/@Table[GraphData[{"Pan",n},"DistanceMatrix"],{n,3,20}]

MatrixForm/@Table[SparseArray[{i_,j_}:>Abs[i-j],n],{n,3,20}]

DistancePolynomial

(poly=Factor[Table[GraphData[{"Pan",n},"DistancePolynomial"],{n,k0=3,kmax=20}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[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,k0,kmax}]

FullSimplify[poly2-poly]

DistinguishingNumber

seq=Table[GraphData[{"Pan",n},"DistinguishingNumber"],{n,3,20}]

DominatingSetCount

seq=Table[GraphData[{"Pan",n},"DominatingSetCount"],{n,3,20}]

FindSequenceFunction[seq,n-2]

FullSimplify[Root[-11-39 #1-55 ((#1)^2)+11 ((#1)^3)&,1]((Root[-1-#1-((#1)^2)+((#1)^3)&,1])^(-2))]

Table[((Root[-1-#1-((#1)^2)+((#1)^3)&,1])^n) Root[-11+31 #1-33 ((#1)^2)+11 ((#1)^3)&,1]+((Root[-1-#1-((#1)^2)+((#1)^3)&,3])^n) Root[-11+31 #1-33 ((#1)^2)+11 ((#1)^3)&,2]+((Root[-1-#1-((#1)^2)+((#1)^3)&,2])^n) Root[-11+31 #1-33 ((#1)^2)+11 ((#1)^3)&,3],{n,3,10}]//FullSimplify

DominationNumber

seq=Table[GraphData[{"Pan",n},"DominationNumber"],{n,3,20}]

DominationPolynomial

(poly=Factor[Table[GraphData[{"Pan",n},"DominationPolynomial"],{n,k0=3,kmax=20}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[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,k0,kmax}]

FullSimplify[poly2-poly]

EdgeChromaticNumber

seq=Table[GraphData[{"Pan",n},"EdgeChromaticNumber"],{n,3,20}]

EdgeConnectivity

Table[GraphData[{"Pan",n},"EdgeConnectivity"],{n,3,20}]

EdgeCount

seq=Table[GraphData[{"Pan",n},"EdgeCount"],{n,3,20}]

FindSequenceFunction[seq,n-2]

EdgeCoverCount

seq=Table[GraphData[{"Pan",n},"EdgeCoverCount"],{n,3,20}]

Table[Fibonacci[n+2],{n,3,20}]

EdgeCoverNumber

seq=Table[GraphData[{"Pan",n},"EdgeCoverNumber"],{n,3,20}]

EdgeCoverPolynomial

(poly=Factor[Table[GraphData[{"Pan",n},"EdgeCoverPolynomial"],{n,k0=3,kmax=20}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[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,k0,kmax}]

FullSimplify[poly2-poly]

EulerianCycleCount

seq=Table[GraphData[{"Pan",n},"EulerianCycleCount"],{n,3,20}]

FaceCount

seq=Table[GraphData[{"Pan",n},"FaceCount"],{n,3,20}]

FlowPolynomial

(poly=Factor[Table[GraphData[{"Pan",n},"FlowPolynomial"],{n,k0=3,kmax=20}]/.f_Function:>f[x]])

FractionalChromaticNumber

seq=Table[GraphData[{"Pan",n},"FractionalChromaticNumber"],{n,3,20}]

FindGeneratingFunction[seq,n-2]

gf=(x^3)FindGeneratingFunction[seq,x]

FullSimplify[SeriesCoefficient[gf,{x,0,n}],n>2]

FractionalEdgeChromaticNumber

seq=Table[GraphData[{"Pan",n},"FractionalEdgeChromaticNumber"],{n,3,20}]

Girth

seq=Table[GraphData[{"Pan",n},"Girth"],{n,3,20}]

Gonality

seq=Table[GraphData[{"Pan",n},"Gonality"],{n,3,20}]

HamiltonConnected

seq=Table[GraphData[{"Pan",n},"HamiltonConnected"],{n,3,20}]

HamiltonDecompositionCount

seq=Table[GraphData[{"Pan",n},"HamiltonDecompositionCount"],{n,3,20}]

Hamiltonian

Table[GraphData[{"Pan",n},"Hamiltonian"],{n,3,20}]

HamiltonianCycleCount

seq=Table[GraphData[{"Pan",n},"HamiltonianCycleCount"],{n,3,20}]

HamiltonianNumber

seq=Table[GraphData[{"Pan",n},"HamiltonianNumber"],{n,3,20}]

FindSequenceFunction[seq,n-2]

HamiltonianPathCount

seq=Table[GraphData[{"Pan",n},"HamiltonianPathCount"],{n,3,20}]

HamiltonianWalkCount

seq=Table[GraphData[{"Pan",n},"HamiltonianWalkCount"],{n,3,20}]

HararyIndex

seq=Table[GraphData[{"Pan",n},"HararyIndex"],{n,3,20}]

FindSequenceFunction[seq,n-2]

FindGeneratingFunction[seq,x]

HexagonCount

seq=Table[GraphData[{"Pan",n},"HexagonCount"],{n,3,20}]

IdiosyncraticPolynomial

(poly=Factor[Table[GraphData[{"Pan",n},"IdiosyncraticPolynomial"][x,y],{n,k0=3,kmax=20}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[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,k0,kmax}]

FullSimplify[poly2-poly]

IndependenceNumber

seq=Table[GraphData[{"Pan",n},"IndependenceNumber"],{n,3,20}]

FindSequenceFunction[seq,n-2]//FullSimplify

IndependencePolynomial

(poly=Factor[Table[GraphData[{"Pan",n},"IndependencePolynomial"],{n,k0=3,kmax=20}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[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,k0,kmax}]

FullSimplify[poly2-poly]

IndependenceRatio

seq=Table[GraphData[{"Pan",n},"IndependenceRatio"],{n,3,20}]

FindSequenceFunction[seq,n-2]//FullSimplify[#,n∈Integers]&//Factor

IndependentEdgeSetCount

A006355

seq=Table[GraphData[{"Pan",n},"IndependentEdgeSetCount"],{n,3,20}]

FindSequenceFunction[seq,n-2]//FullSimplify

Table[2Fibonacci[n+1],{n,0,20}]

Table[2Fibonacci[n+1],{n,3,20}]

Join[{1},Table[2Fibonacci[n],{n,0,20}]]

seq=Join[{1},2Fibonacci[Range[0,20]]]

FindLinearRecurrence[Rest@seq]

seq=LinearRecurrence[{1,1},{6,10},20]

gf=FindGeneratingFunction[seq,x]

CoefficientList[Series[((-1+x-(x^2))/(-1+x+(x^2))),{x,0,20}],x]

IndependentVertexSetCount

A001060

seq=Table[GraphData[{"Pan",n},"IndependentVertexSetCount"],{n,3,20}]

FindSequenceFunction[seq,n-2]

Table[Fibonacci[n+1]+LucasL[n],{n,3,20}]

Table[Fibonacci[n+2]+LucasL[n+1],{n,0,20}]

LinearRecurrence[{1,1},{2,5},{0,10}]

Table[Fibonacci[n+4]-Fibonacci[n-1],{n,0,60}]

FindLinearRecurrence[DeleteMissing@seq]

seq=LinearRecurrence[{1,1},{2,5},50]

gf=x FindGeneratingFunction[DeleteMissing@seq,x]

CoefficientList[Series[((-2-3 x)/(-1+x+(x^2))),{x,0,20}],x]

IntersectionNumber

seq=Table[GraphData[{"Pan",n},"IntersectionNumber"],{n,3,20}]

Pan

A291107

seq=Table[GraphData[{"Pan",n},"IrredundantSetCount"],{n,3,20}]

coef=FindLinearRecurrence[DeleteMissing@seq]

Take[seq,Length@coef]

seq=LinearRecurrence[{1,1,0,1,0,-1},{3,5,7,16,26,42,72,125},40]

FindSequenceFunction[DeleteMissing@seq,n-2]

gf=x FindGeneratingFunction[DeleteMissing@seq,x]

CoefficientList[Series[((3+2 x-(x^2)+4 (x^3)-5 (x^5))/(1-x-(x^2)-(x^4)+(x^6))),{x,0,20}],x]

res=Refine[SeriesCoefficient[gf,{x,0,n}],n>2]

terms=List@@res;

terms[[1]]

N[terms/.n->0]//Chop

approx=RootApproximant[N[exact=terms/.n->0,200],12]

pos=approx[[All,-2]]

N[approx-exact,200]

f1=1-((#1)^2)-((#1)^4)-((#1)^5)+((#1)^6)&;
f2=23827-181313 #1+573482 ((#1)^2)-964066 ((#1)^3)+906011 ((#1)^4)-448265 ((#1)^5)+89653 ((#1)^6)&;

Table[Sum[((Root[f1,k])^n)Root[f2,pos[[k]]],{k,Length[pos]}],{n,3,20}]//N//Chop

an=ToNumberField[Root[f2,pos[[#]]],Root[f1,#]]&/@Range[Length[pos]]

Equal@@Last/@an

RootSum[f1,Evaluate[((#)^n) #^Range[0,Length[pos]-1].an[[1,2]]]&]

Table[((RootSum[1-((#1)^2)-((#1)^4)-((#1)^5)+((#1)^6)&,59603 ((#1)^n)-12384 ((#1)^(1+n))-10700 ((#1)^(2+n))+17668 ((#1)^(3+n))+3457 ((#1)^(4+n))+1652 ((#1)^(5+n))&])/89653),{n,3,20}]

Table[RootSum[1-((#)^2)-((#)^4)-((#)^5)+((#)^6)&,59603 ((#)^n)-12384 ((#)^(1+n))-10700 ((#)^(2+n))+17668 ((#)^(3+n))+3457 ((#)^(4+n))+1652 ((#)^(5+n))&]/89653,{n,20}]

KirchhoffIndex

seq=Table[GraphData[{"Pan",n},"KirchhoffIndex"],{n,3,20}]

FindSequenceFunction[seq,n-2]//Factor

KirchhoffSumIndex

seq=Table[GraphData[{"Pan",n},"KirchhoffSumIndex"],{n,3,20}]

FindSequenceFunction[seq,n-2]//Factor

LaplacianMatrix

MatrixForm/@Table[GraphData[{"Pan",n},"LaplacianMatrix"],{n,3,20}]

LaplacianPolynomial

(poly=Factor[Table[GraphData[{"Pan",n},"LaplacianPolynomial"],{n,k0=3,kmax=20}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[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,k0,kmax}]

FullSimplify[poly2-poly]

LeafCount

seq=Table[GraphData[{"Pan",n},"LeafCount"],{n,3,20}]

LongestCycleCount

seq=Table[GraphData[{"Pan",n},"LongestCycleCount"],{n,3,20}]

LovaszNumber

seq=Table[GraphData[{"Pan",n},"LovaszNumber"],{n,3,20}]

MatchingGeneratingPolynomial

(poly=Factor[Table[GraphData[{"Pan",n},"MatchingGeneratingPolynomial"],{n,k0=3,kmax=20}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[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,k0,kmax}]

FullSimplify[poly2-poly]

MatchingNumber

seq=Table[GraphData[{"Pan",n},"MatchingNumber"],{n,3,20}]

MatchingPolynomial

(poly=Factor[Table[GraphData[{"Pan",n},"MatchingPolynomial"],{n,k0=3,kmax=20}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[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,k0,kmax}]

FullSimplify[poly2-poly]

MaximalCliqueCount

seq=Table[GraphData[{"Pan",n},"MaximalCliqueCount"],{n,3,20}]

FindSequenceFunction[seq,n-2]

SeriesCoefficient[FindGeneratingFunction[seq,x],{x,0,n}]

MaximalIndependentEdgeSetCount

seq=Table[GraphData[{"Pan",n},"MaximalIndependentEdgeSetCount"],{n,3,20}]

FindSequenceFunction[seq,n-2]//FullSimplify

FullSimplify[Root[-7+36 #1-69 ((#1)^2)+23 ((#1)^3)&,1]((Root[-1-#1+((#1)^3)&,1])^(-2))]

MaximalIndependentVertexSetCount

A084338

seq=Table[GraphData[{"Pan",n},"MaximalIndependentVertexSetCount"],{n,3,20}]

FindLinearRecurrence[DeleteMissing@seq]

seq=LinearRecurrence[{0,1,1},{1,2,3},20]

LinearRecurrence[{0,1,1},{3,3,5},20]

FindSequenceFunction[seq,n]

f1=-1-#1+((#1)^3)&;
f2=-5+27 #1-46 ((#1)^2)+23 ((#1)^3)&;
pos={1,3,2};

Table[Sum[((Root[f1,k])^(n-2))Root[f2,pos[[k]]],{k,Length[pos]}],{n,3,10}]//N//Chop

an=ToNumberField[Root[f2,pos[[#]]],Root[f1,#]]&/@Range[Length[pos]]

Equal@@Last/@an

an[[1,2]]

RootSum[f1,Evaluate[((#)^(n+Range[Length[pos]]-1)).an[[1,2]]]&]

Table[RootSum[-1-#+((#)^3)&,12 ((#)^n)+4 ((#)^(n+1))+5 ((#)^(n+2))&]/23,{n,20}]

gf=x FindGeneratingFunction[LinearRecurrence[{0,1,1},{1,2,3},20],x]

CoefficientList[Series[((-1-2 x-2 (x^2))/(-1+(x^2)+(x^3))),{x,0,20}],x]

MaximalIrredundantSetCount

A291102

seq=Table[GraphData[{"Pan",n},"MaximalIrredundantSetCount"],{n,3,20}]

seq={2,2,3,7,9,13,19,24,39,63,87,124,183,272,405,593,867,1261,1869,2760,4046,5936,8712,12817,18861,27720,40711,59792,87915,129250,189946,279118,410135,602803,886008,1302157,1913622,2812220,4133091,6074385,8927330,13119959};

coef=FindLinearRecurrence[DeleteMissing@seq]

Take[seq,Length@coef]

LinearRecurrence[{0,1,1,1,1,0,-1,-2,-1,2,1,0,0,-1},{2,2,3,7,9,13,19,24,39,63,87,124,183,272},20]

FindSequenceFunction[DeleteMissing@seq,n]

gf=x FindGeneratingFunction[DeleteMissing@seq,x]

gf=((x (2+2 x+(x^2)+3 (x^3)+2 (x^4)-(x^5)-2 (x^6)-6 (x^7)-3 (x^8)+7 (x^9)+3 (x^10)-(x^11)-4 (x^13)))/(1-(x^2)-(x^3)-(x^4)-(x^5)+(x^7)+2 (x^8)+(x^9)-2 (x^10)-(x^11)+(x^14)));

gf//Factor

Series[gf,{x,0,10}]

CoefficientList[Series[((2+2 x+(x^2)+3 (x^3)+2 (x^4)-(x^5)-2 (x^6)-6 (x^7)-3 (x^8)+7 (x^9)+3 (x^10)-(x^11)-4 (x^13))/(1-(x^2)-(x^3)-(x^4)-(x^5)+(x^7)+2 (x^8)+(x^9)-2 (x^10)-(x^11)+(x^14))),{x,0,20}],x]

res=Refine[SeriesCoefficient[gf,{x,0,n}],n>0]

terms=List@@res;

Table[k->N[terms/.n->k],{k,-2,2}]//Chop

approx=RootApproximant[N[exact=terms/.n->-1,300],14]

pos=approx[[All,-2]]

N[approx-exact,300]

f1=1-((#1)^3)-2 ((#1)^4)+((#1)^5)+2 ((#1)^6)+((#1)^7)-((#1)^9)-((#1)^10)-((#1)^11)-((#1)^12)+((#1)^14)&;
f2=21906349+144822832 #1+3821131687 ((#1)^2)-33132433375 ((#1)^3)-271071410794 ((#1)^4)-1331747633483 ((#1)^5)+10069218001916 ((#1)^6)+79192919106972 ((#1)^7)+20033982721254 ((#1)^8)-238607135560514 ((#1)^9)-459108934406554 ((#1)^10)-8307005186288589 ((#1)^11)-23521725068876263 ((#1)^12)+47617929706047629 ((#1)^14)&;

Table[Sum[((Root[f1,k])^(n+1))Root[f2,pos[[k]]],{k,Length[pos]}],{n,10}]//N//Chop

(an=ToNumberField[Root[f2,pos[[#]]],Root[f1,#]]&/@Take[Range[Length[pos]],1])//Timing

RootSum[f1,Evaluate[((#)^(n+Range[Length[pos]])).an[[1,2]]]&]

Table[-RootSum[1-#^3-2 #^4+#^5+2 #^6+#^7-#^9-#^10-#^11-#^12+#^14&,500851004670498 #^(n+1)-3689002954543242 #^(n+2)-4674357321032747 #^(n+3)-11682114439256677 #^(n+4)+4235991226348286 #^(n+5)+7038537508218316 #^(n+6)+7181640141870472 #^(n+7)+1546373234795414 #^(n+8)-8648457478830123 #^(n+9)-8135065519248445 #^(n+10)-4540890555566032 #^(n+11)-5314826024895471 #^(n+12)-1546564184442276 #^(n+13)+6933486092556085 #^(n+14)&]/47617929706047629,{n,10}]

MaximumCliqueCount

seq=Table[GraphData[{"Pan",n},"MaximumCliqueCount"],{n,3,20}]

FindSequenceFunction[seq,n-2]

SeriesCoefficient[FindGeneratingFunction[seq,x],{x,0,n}]

MaximumIndependentEdgeSetCount

seq=Table[GraphData[{"Pan",n},"MaximumIndependentEdgeSetCount"],{n,3,20}]

MaximumIndependentVertexSetCount

seq=Table[GraphData[{"Pan",n},"MaximumIndependentVertexSetCount"],{n,3,20}]

MaximumLeafNumber

seq=Table[GraphData[{"Pan",n},"MaximumLeafNumber"],{n,3,20}]

MaximumVertexDegree

seq=Table[GraphData[{"Pan",n},"MaximumVertexDegree"],{n,3,20}]

MeanDistance

seq=Table[GraphData[{"Pan",n},"MeanDistance"],{n,3,20}]

FindSequenceFunction[seq,n-2]//FullSimplify//Factor

MinimalDominatingSetCount

A290273

seq=Table[GraphData[{"Pan",n},"MinimalDominatingSetCount"],{n,3,20}]

FindLinearRecurrence[DeleteMissing@seq]

LinearRecurrence[{0,1,1,1,0,-1},{3,5,7,8,13,18,25},{-1,10}]

seq=LinearRecurrence[{0,1,1,1,0,-1},{2,2,3,5,7,8},50]

FindSequenceFunction[DeleteMissing@seq,n]

gf=x FindGeneratingFunction[DeleteMissing@seq,x]

gf=((x (2+2 x+(x^2)+(x^3)-2 (x^5)))/(1-(x^2)-(x^3)-(x^4)+(x^6)));

ToString[gf,InputForm]

CoefficientList[Series[((2+2 x+(x^2)+(x^3)-2 (x^5))/(1-(x^2)-(x^3)-(x^4)+(x^6))),{x,0,20}],x]

res=Refine[SeriesCoefficient[gf,{x,0,n}],n>0]

res=(((Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,1])^(1+n)) (-1-Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,1]+2 ((Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,1])^3)+2 ((Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,1])^4)))/((Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,1]-Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,2]) (Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,1]-Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,3]) (Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,1]-Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,4]) (Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,1]-Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,5]) (Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,1]-Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,6]))+(((Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,2])^(1+n)) (-1-Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,2]+2 ((Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,2])^3)+2 ((Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,2])^4)))/((-Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,1]+Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,2]) (Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,2]-Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,3]) (Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,2]-Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,4]) (Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,2]-Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,5]) (Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,2]-Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,6]))+(((Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,3])^(1+n)) (-1-Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,3]+2 ((Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,3])^3)+2 ((Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,3])^4)))/((-Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,1]+Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,3]) (-Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,2]+Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,3]) (Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,3]-Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,4]) (Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,3]-Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,5]) (Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,3]-Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,6]))+(((Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,4])^(1+n)) (-1-Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,4]+2 ((Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,4])^3)+2 ((Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,4])^4)))/((-Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,1]+Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,4]) (-Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,2]+Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,4]) (-Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,3]+Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,4]) (Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,4]-Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,5]) (Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,4]-Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,6]))+(((Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,5])^(1+n)) (-1-Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,5]+2 ((Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,5])^3)+2 ((Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,5])^4)))/((-Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,1]+Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,5]) (-Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,2]+Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,5]) (-Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,3]+Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,5]) (-Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,4]+Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,5]) (Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,5]-Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,6]))+(((Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,6])^(1+n)) (-1-Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,6]+2 ((Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,6])^3)+2 ((Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,6])^4)))/((-Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,1]+Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,6]) (-Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,2]+Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,6]) (-Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,3]+Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,6]) (-Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,4]+Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,6]) (-Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,5]+Root[1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&,6]));

terms=List@@res;

terms[[1]]

N[terms/.n->0]//Chop

approx=RootApproximant[N[exact=terms/.n->0,200],6]

pos=approx[[All,-2]]

N[approx-exact,200]

f1=1-((#1)^2)-((#1)^3)-((#1)^4)+((#1)^6)&;
f2=29-643 #1+5541 ((#1)^2)-25877 ((#1)^3)+68700 ((#1)^4)-104882 ((#1)^5)+52441 ((#1)^6)&;

seq

Table[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

an[[1,2]]

RootSum[f1,Evaluate[((#)^n) #^Range[0,Length[pos]-1].an[[1,2]]]&]

Table[-RootSum[1-((#)^2)-((#)^3)-((#)^4)+((#)^6)&,-9 ((#)^n)+33 ((#)^(n+1))-23 ((#)^(n+2))-45 ((#)^(n+3))-38 ((#)^(n+4))+((#)^(n+5))&]/229,{n,20}]

MinimalDominatingSetSignature

seq=Table[GraphData[{"Pan",n},"MinimalDominatingSetSignature"],{n,3,20}]//Column

MinimalEdgeCoverCount

Padovan numbers

A000931

seq=Table[GraphData[{"Pan",n},"MinimalEdgeCoverCount"],{n,3,20}]

FindLinearRecurrence[DeleteMissing@seq]

FindSequenceFunction[{1,0,0,1,0,1,1,1,2,2,3,4,5,7,9,12,16,21,28,37,49,65},n]

Table[RootSum[-1-#+((#)^3)&,-11 ((#)^(n+7))+4 ((#)^(n+8))+5 ((#)^(n+9))&]/23,{n,3,20}]

MinimalEdgeCoverSignature

seq=Table[GraphData[{"Pan",n},"MinimalEdgeCoverSignature"],{n,3,20}]//Column

MinimumCliqueCoveringCount

seq=Table[GraphData[{"Pan",n},"MinimumCliqueCoveringCount"],{n,3,20}]

Table[Piecewise[{{1+n/2,Mod[n,2]==0},{1,Mod[n,2]==1}}],{n,3,20}]

MinimumConnectedDominatingSetCount

seq=Table[GraphData[{"Pan",n},"MinimumConnectedDominatingSetCount"],{n,3,20}]

FindSequenceFunction[seq,n-2]

MinimumDistinguishingLabelingCount

seq=Table[GraphData[{"Pan",n},"MinimumDistinguishingLabelingCount"],{n,3,20}]

FindSequenceFunction[seq,n-2]//FullSimplify

Table[-(2^(n/2))(-2-(Sqrt[2])+(((-1))^n) (-2+(Sqrt[2]))),{n,3,10}]

Table[(2^(n+1))-(2^(2+Floor[n/2])),{n,3,20}]

FullSimplify[(2^(n+1))-(2^(2+Floor[n/2])),Mod[n,2]==#]&/@{0,1}

MinimumDominatingSetCount

seq=Table[GraphData[{"Pan",n},"MinimumDominatingSetCount"],{n,3,20}]

MinimumEdgeCoverCount

seq=Table[GraphData[{"Pan",n},"MinimumEdgeCoverCount"],{n,3,20}]

MinimumVertexDegree

seq=Table[GraphData[{"Pan",n},"MinimumVertexDegree"],{n,3,20}]

MolecularTopologicalIndex

seq=Table[GraphData[{"Pan",n},"MolecularTopologicalIndex"],{n,3,20}]

FindSequenceFunction[seq,n-2]//FullSimplify//Factor

gf=(x^2)FindGeneratingFunction[seq,x]//Factor

Nonhamiltonian

Table[GraphData[{"Pan",n},"Nonhamiltonian"],{n,3,20}]

NormalizedLaplacianMatrix

MatrixForm/@Table[GraphData[{"Pan",n},"NormalizedLaplacianMatrix"],{n,3,20}]

OddChordlessCycleCount

seq=Table[GraphData[{"Pan",n},"OddChordlessCycleCount"],{n,3,20}]

PathCount

seq=Table[GraphData[{"Pan",n},"PathCount"],{n,3,20}]

FindLinearRecurrence[seq]

FindSequenceFunction[seq,n-2]//Factor

gf=x FindGeneratingFunction[seq,x]

FullSimplify[SeriesCoefficient[gf,{x,0,n}],n>0]//Factor

PathPolynomial

(poly=Factor[Table[GraphData[{"Pan",n},"PathPolynomial"],{n,k0=3,kmax=20}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[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,k0,kmax}]

FullSimplify[poly2-poly]

PentagonCount

seq=Table[GraphData[{"Pan",n},"PentagonCount"],{n,3,20}]

Perfect

Table[GraphData[{"Pan",n},"Perfect"],{n,3,20}]

ProjectivePlaneCrossingNumber

seq=Table[GraphData[{"Pan",n},"ProjectivePlaneCrossingNumber"],{n,3,20}]

Radius

seq=Table[GraphData[{"Pan",n},"Radius"],{n,3,20}]

RankPolynomial

(poly=Factor[Table[GraphData[{"Pan",n},"RankPolynomial"][x,y],{n,k0=3,kmax=20}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[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,k0,kmax}]

FullSimplify[poly2-poly]

RectilinearCrossingNumber

seq=Table[GraphData[{"Pan",n},"RectilinearCrossingNumber"],{n,3,20}]

ReliabilityPolynomial

(poly=Factor[Table[GraphData[{"Pan",n},"ReliabilityPolynomial"],{n,k0=3,kmax=20}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[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,k0,kmax}]

FullSimplify[poly2-poly]

SigmaPolynomial

(poly=Factor[Table[GraphData[{"Pan",n},"SigmaPolynomial"],{n,k0=3,kmax=20}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[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,k0,kmax}]

FullSimplify[poly2-poly]

Skewness

GraphData[{"Pan",6}]

seq=Table[GraphData[{"Pan",n},"Skewness"],{n,3,20}]

SpanningTreeCount

seq=Table[GraphData[{"Pan",n},"SpanningTreeCount"],{n,3,20}]

SquareCount

seq=Table[GraphData[{"Pan",n},"SquareCount"],{n,3,20}]

StabilityIndex

seq=Table[GraphData[{"Pan",n},"StabilityIndex"],{n,3,20}]

FindSequenceFunction[seq,n-2]//FullSimplify[#,n∈Integers]&

gf=(x^3)FindGeneratingFunction[seq,x]

TopologicalIndex

seq=Table[GraphData[{"Pan",n},"TopologicalIndex"],{n,3,20}]

FindSequenceFunction[seq,n-2]

FindGeneratingFunction[seq,x]

ToroidalCrossingNumber

seq=Table[GraphData[{"Pan",n},"ToroidalCrossingNumber"],{n,3,20}]

Traceable

Table[GraphData[{"Pan",n},"Traceable"],{n,3,20}]

TriangleCount

seq=Table[GraphData[{"Pan",n},"TriangleCount"],{n,3,20}]

TuttePolynomial

(poly=Factor[Table[GraphData[{"Pan",n},"TuttePolynomial"][x,y],{n,k0=3,kmax=20}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[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,k0,kmax}]

FullSimplify[poly2-poly]

Untraceable

Table[GraphData[{"Pan",n},"Untraceable"],{n,3,20}]

VertexConnectivity

Table[GraphData[{"Pan",n},"VertexConnectivity"],{n,3,20}]

VertexCount

seq=Table[GraphData[{"Pan",n},"VertexCount"],{n,3,20}]

FindSequenceFunction[seq,n-2]

VertexCoverNumber

vertex cover number == vertex count - independence number

seq=Table[GraphData[{"Pan",n},"VertexCoverNumber"],{n,3,20}]

FindSequenceFunction[seq,n-2]//FullSimplify

gf=x FindGeneratingFunction[seq,x]

FullSimplify[SeriesCoefficient[gf,{x,0,n}],n>0]

VertexCoverPolynomial

(poly=Factor[Table[GraphData[{"Pan",n},"VertexCoverPolynomial"],{n,k0=3,kmax=20}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[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,k0,kmax}]

FullSimplify[poly2-poly]

VertexDegrees

seq=Table[GraphData[{"Pan",n},"VertexDegrees"],{n,3,20}]

VertexTransitive

Table[GraphData[{"Pan",n},"VertexTransitive"],{n,3,20}]

WienerIndex

A180861

seq=Table[GraphData[{"Pan",n},"WienerIndex"],{n,3,20}]

FindSequenceFunction[seq,n-2]//FullSimplify

Table[(-2+15 n+4 (n^2)+2 (n^3)+(((-1))^n) (2+n))/16,{n,20}]

FindLinearRecurrence[seq]

LinearRecurrence[{2,1,-4,1,2,-1},{8,16,26,42,61,88},{-1,1}]

seq=LinearRecurrence[{2,1,-4,1,2,-1},{1,4,8,16,26,42},20]

gf=x FindGeneratingFunction[seq,x]

CoefficientList[Series[((1+2 x-(x^2)+(x^4))/((((-1+x))^4) (((1+x))^2))),{x,0,20}],x]

WienerSumIndex

seq=Table[GraphData[{"Pan",n},"WienerSumIndex"],{n,3,20}]

FindSequenceFunction[seq,n-2]

GraphData

ClassString

GraphClassString[GraphData["Pan"]]

GraphDataString

Do[Print[GraphDataString[
PanGraph[n],
"StandardName"->{"Pan",n},
"AlternateStandardName"->{"Tadpole",{n,1}},
"Name"->ToString[n]<>"-pan graph",
"AlternateNames"->"("<>ToString[n]<>",1)-tadpole graph",
"Classes"->{"Pan","Tadpole"},
"NotationRules"->{"Pan"->n,"Tadpole"->{n,1}},
TimeConstraint->300,
Debug->True
]],{n,8,20}]

Add UnitDistance

GraphData[{"Pan",5}]

l=Table[{gn={"Pan",n},{0,ToCommonEdges[PanGraph[n,GraphLayout->#],System`Graph[GraphData[gn,"Vertices"],GraphData[gn,"Edges"]]]&/@{Automatic,"UnitDistance"}//Simplify}},{n,5,20}]

PropertyReplace[l,"Embeddings","11.3"]

RenameGraphFiles[]

l=Table[{gn={"Pan",n},{0,{{"MinimalCrossing","MinimalRectilinearCrossing","Planar","Primary",},{"Integral","MinimalCrossing","MinimalIntegral","MinimalPlanarIntegral","MinimalRectilinearCrossing","Planar","UnitDistance"}}}},{n,5,20}]

PropertyReplace[l,"EmbeddingClasses","11.3"]

RenameGraphFiles[]