Total Dominating Set

Author

Eric W. Weisstein
February 9, 2024

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

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

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

Figure

HighlightGraph[GraphData["P","Graph","Labeled"],GraphData["P","TotalDominatingSets"][[2]],VertexSize->.12,ImageSize->300]

Parametrized families

GraphDataCountTable["TotalDominatingSetCount",{"Alkane"->347478,"Andrasfai"->302759,"Antiprism"->302760,"Apollonian"->347479,"Barbell"->302761,"Bishop"->304555,"BlackBishop"->303145,"Book"->028400,"Bruhat"->323336,"Centipede"->151821,"CocktailParty"->289254,"Complete"->000295,"CompleteBipartite"->060867,"CompleteTripartite"->302942,"CrossedPrism"->302941,"Crown"->294140,"Cycle"->001638,"CycleComplement"->347477,"DiagonalIntersection"->347480,"Dipyramid"->347481,"DoubleCone"->362750,"Empty"->000007,"FibonacciCube"->303047,"FoldedCube"->347482,"Gear"->303834,{"Grid",2}->133793,{"Grid",3}->303897,"HalvedCube"->347483,"Hanoi"->303898,"Helm"->000079,"Hypercube"->295372,"King"->303116,"Knight"->303155,"Ladder"->141583,"LadderRung"->000012,"LucasCube"->365572,"MoebiusLadder"->295420,"Mycielski"->303067,"Pan"->302506,"Path"->195971,"PathComplement"->302689,"Pell"->365571,"Prism"->296102,"Queen"->303207,"Rook"->303208,"RookComplement"->303209,"SierpinskiSieve"->347484,"Star"->000225,"Sun"->087131,"Sunlet"->000079,"TorusGrid"->303210,"Transposition"->303146,"Trapezohedron"->362751,"Triangular"->303048,"TriangularGrid"->303049,"TriangularHoneycombAcuteKnight"->304552,"TriangularHoneycombBishop"->304553,"TriangularHoneycombObtuseKnight"->304554,"TriangularHoneycombQueen"->304556,"TriangularHoneycombRook"->000007,"TriangularSnake"->308592,"Web"->000302,"Wheel"->302603,"WhiteBishop"->303147}]

Parametrized families

<<MathWorld`Graphs`

Andrasfai

A302759

seq=Table[GraphData[{"Andrasfai",n},"TotalDominatingSetCount"],{n,10}]

seq={1,11,131,1365,12883,113935,967455,8013983,65410751,529283583,4261449727,34213027327,274240586751,2196272295935,17580376055807,140687025184767,1125685164621823,9006288735567871,72053745778425855,576444534576513023,4611617848860868607};

FindSequenceFunction[DeleteMissing@seq,n]//Head//Timing

coef=FindLinearRecurrence[DeleteMissing@Rest@seq]

Length@coef

Take[Rest@seq,Length@coef]

Join[{1},LinearRecurrence[{23,-210,996,-2664,4032,-3200,1024},{11,131,1365,12883,113935,967455,8013983},20]]

gf=x FindGeneratingFunction[DeleteMissing@seq,x]

InputForm[gf]

CoefficientList[Series[((-1+12 x-88 (x^2)+334 (x^3)-706 (x^4)+928 (x^5)-672 (x^6)+256 (x^7))/((((-1+2 x))^3) (((-1+4 x))^2) (1-9 x+8 (x^2)))),{x,0,20}],x]

FullSimplify[SeriesCoefficient[gf,{x,0,n}],n>1]

Join[{1},Table[((8^(n+1))+((2^n)(n-2)-(4^(n+1)))(3 n-1))/16-1,{n,2,20}]]

Alkane

seq=Table[GraphData[{"Alkane",n},"TotalDominatingSetCount"],{n,10}]

FindSequenceFunction[Rest@seq,n-1]

Table[Piecewise[{{15,n==1}},(4^(n+1))],{n,30}]

Antiprism

A302760

Entity["Graph",{"Antiprism",n}]["TotalDominatingSetCount"]

seq=Table[GraphData[{"Antiprism",n},"TotalDominatingSetCount"],{n,3,20}]

seq={3,11,54,179,648,2414,8809,32195,117945,431696,1579955,5783294,21168592,77482521,283608249,1038086883,3799689944,13907938601,50906985592,186333942984,682034858839,2496440225499,9137676323347,33446476209566,122423549667123};

FindSequenceFunction[seq,n]

coef=FindLinearRecurrence[seq]

Take[seq,Length[coef]]

LinearRecurrence[{3,1,6,-3,0,0,1},{3,11,54,179,648,2414,8809},20]

gf=x FindGeneratingFunction[DeleteMissing@seq,x]

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

RootReduce[res]

terms=List@@res;

terms[[1]]

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

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

Table[RootSum[-1+3 ((#)^3)-6 ((#)^4)-((#)^5)-3 ((#)^6)+((#)^7)&,((#)^n)&],{n,20}]

RootSum[-1+3 ((#)^3)-6 ((#)^4)-((#)^5)-3 ((#)^6)+((#)^7)&,((#)^(Range[20]))&]

Apollonian

seq=Table[GraphData[{"Apollonian",n},"TotalDominatingSetCount"],{n,6}]

Barbell

A302761

seq=Table[GraphData[{"Barbell",n},"TotalDominatingSetCount"],{n,3,20}]

seq=Table[(2^n-n)^2-(2^n-2*n),{n,20}]

FindSequenceFunction[DeleteMissing@seq,n]//FullSimplify

coef=FindLinearRecurrence[DeleteMissing@seq]

Length@coef

Take[seq,6]

LinearRecurrence[{11,-47,101,-116,68,-16},{1,4,23,136,707,3312},20]

FindGeneratingFunction[DeleteMissing@seq,x]

CoefficientList[Series[((1-7 x+26 (x^2)-30 (x^3)+4 (x^4))/((((-1+x))^3) (((-1+2 x))^2) (-1+4 x))),{x,0,20}],x]

Bishop n×n

A304555

seq=Table[GraphData[{"Bishop",{n,n}},"TotalDominatingSetCount"],{n,10}]

seq={0,1,144,20449,14758132,39638819025,396238966098460,14867893869236421601,2126342662340665160447328,1176421647617108793775585682449};

FindSequenceFunction[DeleteMissing@Drop[seq,1],n-1]//Head//Timing

FindLinearRecurrence[Drop[seq,2]]//Head//Timing

FindGeneratingFunction[seq,x]//Head//Timing

BlackBishop n×n

A303145

seq=Table[GraphData[{"BlackBishop",{n,n}},"TotalDominatingSetCount"],{n,10}]

FindSequenceFunction[DeleteMissing@seq,n]

FindGeneratingFunction[DeleteMissing@seq,x]

Book

seq=Table[GraphData[{"Book",n},"TotalDominatingSetCount"],{n,20}]

FindSequenceFunction[DeleteMissing@seq,n]//FullSimplify

Bruhat

A323336

seq=Table[GraphData[{"Bruhat",n},"TotalDominatingSetCount"],{n,6}]

FindSequenceFunction[DeleteMissing@seq,n]//FullSimplify

Centipede

seq=Table[GraphData[{"Centipede",n},"TotalDominatingSetCount"],{n,10}]

FindSequenceFunction[DeleteMissing@Rest@seq,n-1]

CocktailParty

seq=Table[GraphData[{"CocktailParty",n},"TotalDominatingSetCount"],{n,20}]

FindSequenceFunction[DeleteMissing@seq,n]

Complete

Entity["Graph",{"Complete",n}]["TotalDominatingSetCount"]

seq=Table[GraphData[{"Complete",n},"TotalDominatingSetCount"],{n,20}]

FindSequenceFunction[DeleteMissing@seq,n]

CompleteBipartite m×n

Entity["Graph",{"CompleteBipartite",{m,n}}]["TotalDominatingSetCount"]

Table[GraphData[{"CompleteBipartite",{m,n}},"TotalDominatingSetCount"],{n,10},{m,n}]//Column

Table[((2^m)-1)((2^n)-1),{n,10},{m,n}]//Column

CompleteBipartite n×n

Entity["Graph",{"CompleteBipartite",{n,n}}]["TotalDominatingSetCount"]

seq=Table[GraphData[{"CompleteBipartite",{n,n}},"TotalDominatingSetCount"],{n,20}]

FindSequenceFunction[DeleteMissing@seq,n]//FullSimplify

CompleteTripartite k×m×n

Entity["Graph",{"CompleteTripartite",{k,m,n}}]["TotalDominatingSetCount"]

2-(((1+x))^k)-(((1+x))^m)-(((1+x))^n)+(((1+x))^(k+m+n))/.x->1//FullSimplify

DeleteCases[Flatten[Table[{{k,m,n},GraphData[{"CompleteTripartite",{k,m,n}},"TotalDominatingSetCount"]/.f_Function:>f[x],Expand[2-(2^k)-(2^m)-(2^n)+(2^(k+m+n))]},{n,10},{m,n},{k,m}],2],{_,x_,x_}]

CompleteTripartite n×n×n

A302942

Entity["Graph",{"CompleteTripartite",{n,n,n}}]["TotalDominatingSetCount"]

seq=Table[GraphData[{"CompleteTripartite",{n,n,n}},"TotalDominatingSetCount"],{n,10}]

FindSequenceFunction[DeleteMissing@seq,n]

seq=Table[((((2^n)-1))^2) ((2^n)+2),{n,0,30}]

FindLinearRecurrence[DeleteMissing@seq]

LinearRecurrence[{11,-26,16},{4,54,490},{0,20}]

gf=FindGeneratingFunction[DeleteMissing@seq,x]

InputForm[gf]

CoefficientList[Series[-((2x (2+5 x))/(-1+11 x-26 (x^2)+16 (x^3))),{x,0,20}],x]

CrossedPrism

A302941

seq=Table[GraphData[{"CrossedPrism",2n},"TotalDominatingSetCount"],{n,2,10}]

seq={9,121,1296,14161,154449,1684804,18378369,200477281,2186871696,23855111401,260219353689,2838557779204,30963916217529,337764520613641,3684445810532496,40191139395243841,438418087537149729,4782407823513403204,52168067971110285489};

FindSequenceFunction[DeleteMissing[seq],n]//FullSimplify

Table[2 (((-1))^n)+((((1/2) (11-3 (Sqrt[13]))))^n)+((((1/2) (11+3 (Sqrt[13]))))^n),{n,20}]//FullSimplify

FindLinearRecurrence[seq]

LinearRecurrence[{10,10,-1},{9,121,1296},20]

FindGeneratingFunction[DeleteMissing@seq,x]

CoefficientList[Series[((9+31 x-4 (x^2))/(1-10 x-10 (x^2)+(x^3))),{x,0,20}],x]

Table[((LucasL[n,3])^2),{n,20}]

((LucasL[Range[20],3])^2)

Sqrt[%]

LucasL[Range[20],3]

Crown

A294140

seq=Table[GraphData[{"Crown",n},"TotalDominatingSetCount"],{n,3,20}]

seq=Table[(2^n-1-n)^2,{n,20}]

FindSequenceFunction[DeleteMissing@seq,n]//FullSimplify

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

coef=FindLinearRecurrence[DeleteMissing@seq]

Length@coef

Take[seq,Length@coef]

LinearRecurrence[{11,-47,101,-116,68,-16},{0,1,16,121,676,3249},20]

gf=x FindGeneratingFunction[DeleteMissing@seq,x]

InputForm[gf]

CoefficientList[Series[((x (1+5 x-8 (x^2)-4 (x^3)))/((((-1+x))^3) (((-1+2 x))^2) (-1+4 x))),{x,0,20}],x]

CubeConnectedCycle

Entity["Graph",{"CubeConnectedCycle",n}]["TotalDominatingSetCount"]

l=Table[GraphData[{"CubeConnectedCycle",n},"TotalDominatingSetCount"],{n,3,8}]

Cycle

A001638

Entity["Graph",{"Cycle",n}]["TotalDominatingSetCount"]

seq=Table[GraphData[{"Cycle",n},"TotalDominatingSetCount"],{n,3,20}]

FullSimplify[FindSequenceFunction[DeleteMissing@seq,n-2],n∈Integers]

seq=Table[LucasL[n]+2Cos[n π/2],{n,0,20}]

Table[LucasL[n]+I^n+(-I)^n,{n,0,20}]

FullSimplify[+I^n+(-I)^n]

FindLinearRecurrence[seq]

FindGeneratingFunction[seq,x]

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

CycleComplement

A347477

seq=Table[GraphData[{"CycleComplement",n},"TotalDominatingSetCount"],{n,3,20}]

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

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

Table[Piecewise[{{0,n==3},{1,n==4}},(2^n)-4 n-1],{n,3,40}]

DiagonalIntersection

seq=Table[GraphData[{"DiagonalIntersection",n},"TotalDominatingSetCount"],{n,3,14}]

FindSequenceFunction[DeleteMissing@seq,n-2]

Dipyramid

A347481

seq=Table[GraphData[{"Dipyramid",n},"TotalDominatingSetCount"],{n,3,20}]

FullSimplify[FindSequenceFunction[DeleteMissing@seq,n-2],n∈Integers]

seq=Table[LucasL[n]+2 Cos[((n π)/2)]+3 (2^n)-3,{n,50}]

StringTake[StringTake[ToString[seq],{2,-2}],260]

FindLinearRecurrence[seq]

LinearRecurrence[{4,-5,3,-2,-1,2},{4,10,25,54,104,205},20]

gf=x FindGeneratingFunction[seq,x]//Factor

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

DoubleCone

A362750

seq=Table[GraphData[{"DoubleCone",n},"TotalDominatingSetCount"],{n,3,20}]

FindSequenceFunction[DeleteMissing@seq,n-2]//FullSimplify

A005248

Table[((((1/2) (3-(Sqrt[5]))))^n)+((((1/2) (3+(Sqrt[5]))))^n),{n,10}]//Expand

Table[LucasL[2n],{n,10}]

Table[1+4 (((-1))^n)+(4^n)+LucasL[2n]+4LucasL[n] Cos[((n π)/2)],{n,20}]

FullSimplify[1+4 (((-1))^n)+(4^n)+LucasL[2n]+4LucasL[n] Cos[((n π)/2)]]

FindLinearRecurrence[DeleteMissing@seq]//Head//Timing

gf=(x^3) FindGeneratingFunction[DeleteMissing@seq,x]//FullSimplify

seq=Table[(Fibonacci[n+1]+Fibonacci[n-1]+I^n+(-I)^n)^2+4^n-1,{n,20}]

FullSimplify[(Fibonacci[n+1]+Fibonacci[n-1]+I^n+(-I)^n)^2+4^n-1]

Table[(((2 Cos[((n π)/2)]+Fibonacci[n+1]+Fibonacci[n-1]))^2)+(4^n)-1,{n,20}]

coef=FindLinearRecurrence[seq]

Length[coef]

LinearRecurrence[{7,-15,18,-24,-6,27,-15,13,-4},{4,16,79,336,1144,4351,17224,67936,267919},20]

gf=x FindGeneratingFunction[seq,x]//Factor

FullSimplify[(gf/x)]

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

Empty

Entity["Graph",{"Empty",n}]["TotalDominatingSetCount"]

seq=Table[GraphData[{"Empty",n},"TotalDominatingSetCount"],{n,30}]

FibonacciCube

A303047

seq=Table[GraphData[{"FibonacciCube",n},"TotalDominatingSetCount"],{n,14}]

seq={1,3,12,80,2132,473712,3086726720,5004532472869032};

FindSequenceFunction[seq,n]//Head//Timing

FindLinearRecurrence[seq]//Head//Timing

FindGeneratingFunction[seq,x]//Head//Timing

FoldedCube

seq=Table[GraphData[{"FoldedCube",n},"TotalDominatingSetCount"],{n,2,10}]

Gear

A303834

seq=Table[GraphData[{"Gear",n},"TotalDominatingSetCount"],{n,3,20}]

coef=FindLinearRecurrence[DeleteMissing@seq]

Take[seq,Length@coef]

LinearRecurrence[{4,2,-13,-6,4},{48,161,473,1476,4553},{-1,20}]

seq=LinearRecurrence[{4,2,-13,-6,4},{3,21,48,161,473},20]

FullSimplify[FindSequenceFunction[DeleteMissing@seq,n],n>0&&n∈Integers]

Table[(((1-(Sqrt[5])))^n)+(((1+(Sqrt[5])))^n),{n,10}]//FullSimplify

Table[((((1/2) (3-(Sqrt[5]))))^n)+((((1/2) (3+(Sqrt[5]))))^n),{n,10}]//FullSimplify

FullSimplify[2(((-1))^n)+(2^n)LucasL[n]+LucasL[2n]]

Table[2(((-1))^n)+(2^n)LucasL[n]+LucasL[2n],{n,40}]

gf=x FindGeneratingFunction[DeleteMissing@seq,x,ValidationLength->5]

InputForm[gf]

SeriesCoefficient[((x (-3-9 x+42 (x^2)+34 (x^3)-24 (x^4)))/(-1+4 x+2 (x^2)-13 (x^3)-6 (x^4)+4 (x^5))),{x,0,n}]

CoefficientList[Series[((-3-9 x+42 (x^2)+34 (x^3)-24 (x^4))/(-1+4 x+2 (x^2)-13 (x^3)-6 (x^4)+4 (x^5))),{x,0,20}],x]

Grid m×n

A303111

Entity["Graph",{"Grid",{m,n}}]["TotalDominatingSetCount"]

seq=Table[GraphData[{"Grid",{m,n}},"TotalDominatingSetCount"],{n,10},{m,n}]//Column//TraditionalForm

data=Import["https://oeis.org/A303111/b303111.txt","Table"][[;;,2]];

Quiet@Table[data[[Binomial[i,2]+(1/2) (j-1) (2 i+j)+1]],{i,10},{j,i}]/._Part:>Missing["NotAvailable"]//Column//TraditionalForm

Grid n×n

A133793

Entity["Graph",{"Grid",{n,n}}]["TotalDominatingSetCount"]

seq=Table[GraphData[{"Grid",{n,n}},"TotalDominatingSetCount"],{n,10}]

seq={9,161,11236,3270375,3416753209,13057656650476,183641258024619409,9467975365082755623313,1790666611784474236947695716,1242439796942800308861023912760955};

FindSequenceFunction[seq,n]//Head//Timing

FindGeneratingFunction[seq,x]//Head//Timing

Grid k×m×n

Entity["Graph",{"Grid",{k,m,n}}]["TotalDominatingSetCount"]

seq=Table[GraphData[{"Grid",{k,m,n}},"TotalDominatingSetCount"],{n,10}]

Grid n×n×n

A303897

Entity["Graph",{"Grid",{n,n,n}}]["TotalDominatingSetCount"]

seq=Table[GraphData[{"Grid",{n,n,n}},"TotalDominatingSetCount"],{n,10}]

Sqrt[DeleteMissing@seq]

HalvedCube

A347483

seq=Table[GraphData[{"HalvedCube",n},"TotalDominatingSetCount"],{n,10}]

FindSequenceFunction[DeleteMissing@Rest@seq,n-1]//Head//Timing

FindLinearRecurrence[DeleteMissing@Rest@seq]//Head//Timing

FindGeneratingFunction[DeleteMissing@seq,x]//Head//Timing

Hanoi

A303898

seq=Table[GraphData[{"Hanoi",n},"TotalDominatingSetCount"],{n,5}]

Helm

seq=Table[GraphData[{"Helm",n},"TotalDominatingSetCount"],{n,3,20}]

FindSequenceFunction[DeleteMissing@seq,n-2]

seq=Table[GraphData[{"Hypercube",n},"TotalDominatingSetCount"],{n,10}]

Hypercube

A295372

Entity["Graph",{"Hypercube",n}]["TotalDominatingSetCount"]

seq=Table[GraphData[{"Hypercube",n},"TotalDominatingSetCount"],{n,10}]

Table[Count[Subsets[Range[2^n]],_?(CountDistinct[Flatten[#/.Table[k->AdjacencyList[HypercubeGraph[n],k],{k,2^n}]]]==2^n&)],{n,0,3}]

FindSequenceFunction[DeleteMissing@seq,n]

Keller

Entity["Graph",{"Keller",n}]["TotalDominatingSetCount"]

seq=Table[GraphData[{"Keller",n},"TotalDominatingSetCount"],{n,5}]

King m×n

A303114

Table[GraphData[{"King",{m,n}},"TotalDominatingSetCount"],{n,10},{m,n}]//Column//TraditionalForm

data=Import["https://oeis.org/A303114/b303114.txt","Table"][[2;;,2]]

Quiet@Table[data[[Binomial[i,2]+(1/2) (j-1) (2 i+j)+1]],{i,10},{j,i}]/._Part:>Missing["NotAvailable"]//Column//TraditionalForm

King n×n

A303116

seq=Table[GraphData[{"King",{n,n}},"TotalDominatingSetCount"],{n,20}]

FindSequenceFunction[seq,n]//Head//Timing

FindGeneratingFunction[seq,x]//Head//Timing

Knight m×n

Entity["Graph",{"Knight",{m,n}}]["TotalDominatingSetCount"]

Knight n×n

A303155

Entity["Graph",{"Knight",{n,n}}]["TotalDominatingSetCount"]

seq=Table[GraphData[{"Knight",{n,n}},"TotalDominatingSetCount"],{n,20}]

FindLinearRecurrence[DeleteMissing@seq]//Head//Timing

FindGeneratingFunction[seq,x]//Head//Timing

Ladder

A141583

tribonacci numbers squared

Entity["Graph",{"Ladder",n}]["TotalDominatingSetCount"]

seq=Table[GraphData[{"Ladder",n},"TotalDominatingSetCount"],{n,20}]

Table[((RootSum[-1-#-((#)^2)+((#)^3)&,2 ((#)^(n+1))-4 ((#)^(n+2))+3 ((#)^(n+3))&])^2)/121,{n,20}]

FindLinearRecurrence[DeleteMissing@seq]

FindSequenceFunction[DeleteMissing@seq,n]

FindGeneratingFunction[DeleteMissing@seq,n]

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

seq=Table[((RootSum[-1-#-((#)^2)+((#)^3)&,2 ((#)^n)-4 ((#)^(n+1))+3 ((#)^(n+2))&])^2)/121,{n,0,20}]

FindLinearRecurrence[seq]

LinearRecurrence[{2,3,6,-1,0,-1},{1,1,9,25,81,289},{0,20}]

LadderRung

seq=Table[GraphData[{"LadderRung",n},"TotalDominatingSetCount"],{n,20}]

FindSequenceFunction[seq,n]

LucasCube

A365572

seq=Table[GraphData[{"LucasCube",n},"TotalDominatingSetCount"],{n,15}]

FindLinearRecurrence[DeleteMissing@seq]//Head//Timing

FindGeneratingFunction[DeleteMissing@seq,x]//Head//Timing

MengerSponge

seq=Table[GraphData[{"MengerSponge",n},"TotalDominatingSetCount"],{n,3}]

MoebiusLadder

A295420

Entity["Graph",{"MoebiusLadder",n}]["TotalDominatingSetCount"]

seq=Table[GraphData[{"MoebiusLadder",n},"TotalDominatingSetCount"],{n,3,20}]

seq={1,11,49,131,441,1499,5041,17155,58081,196331,664225,2246915,7601049,25714875,86992929,294294531,995591809,3368061131,11394068049,38545859971,130399709881,441139059867,1492362754129,5048627019523,17079382863841,57779138374059};

FindSequenceFunction[DeleteMissing@seq,n]

coef=FindLinearRecurrence[DeleteMissing@seq]

Length@coef

Take[seq,Length@coef]

LinearRecurrence[{3,0,4,-2,10,4,0,-1,-1},{1,11,49,131,441,1499,5041,17155,58081},20]

gf=x FindGeneratingFunction[DeleteMissing@seq,x]

CoefficientList[Series[((1+8 x+16 (x^2)-20 (x^3)+6 (x^4)-8 (x^5)+4 (x^6)-4 (x^7)-3 (x^8))/(1-3 x-4 (x^3)+2 (x^4)-10 (x^5)-4 (x^6)+(x^8)+(x^9))),{x,0,20}],x]

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

Table[RootSum[-1-#-3 ((#)^2)+((#)^3)&,#^n&]-RootSum[1+#-((#)^2)+((#)^3)&,#^n&]+RootSum[-1+#+((#)^2)+((#)^3)&,#^n&],{n,20}]

{1,-1,1}.#&/@Transpose[RootSum[#,#^Range[20]&]&/@{-1-#-3 ((#)^2)+((#)^3)&,1+#-((#)^2)+((#)^3)&,-1+#+((#)^2)+((#)^3)&}]

Table[RootSum[-1-#-3 ((#)^2)+((#)^3)&,#^n&]-RootSum[1+#-((#)^2)+((#)^3)&,#^n&]+RootSum[-1+#+((#)^2)+((#)^3)&,#^n&],{n,20}]

Mycielski

A303067

related to ConnectedDominatingSetCount

Entity["Graph",{"Mycielski",n}]["TotalDominatingSetCount"]

seq=Table[GraphData[{"Mycielski",n},"TotalDominatingSetCount"],{n,11}]

Odd

Entity["Graph",{"Odd",n}]["TotalDominatingSetCount"]

seq=Table[GraphData[{"Odd",n},"TotalDominatingSetCount"],{n,7}]

Pan

A302506

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

FullSimplify[FindSequenceFunction[DeleteMissing@seq,n-2],n∈Integers]

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

FullSimplify[SeriesCoefficient[gf,{x,0,n}]//ComplexExpand,n>2&&n∈Integers]

Table[(2^(-1-n)) (-3 (((1-(Sqrt[5])))^n) (-3+(Sqrt[5]))+3 (((1+(Sqrt[5])))^n) (3+(Sqrt[5]))),{n,20}]//FullSimplify

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

FullSimplify[(1/5) (2^(-1-n)) ((2^(n+1))3 LucasL[n+2]+3 (2^(2+n)) Cos[((n π)/2)]-(2^(2+n)) Sin[((n π)/2)])]

seq=Table[(3 LucasL[n+2]+6 Cos[((n π)/2)]-2 Sin[((n π)/2)])/5,{n,50}]

FindLinearRecurrence[seq]

LinearRecurrence[{1,0,1,1},{2,3,7,12},20]

gf=x FindGeneratingFunction[seq,x]

InputForm[gf]

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

Path

A195971

Entity["Graph",{"Path",n}]["TotalDominatingSetCount"]

seq=Table[GraphData[{"Path",n},"TotalDominatingSetCount"],{n,20}]

FullSimplify[FindSequenceFunction[DeleteMissing@seq,n],n∈Integers&&n>0]

gf=FindGeneratingFunction[DeleteMissing@seq,x]

FullSimplify[SeriesCoefficient[gf,{x,0,n}]//ComplexExpand,n∈Integers&&n>0]

Table[-((((1/2) (1-(Sqrt[5]))))^n) (-2+(Sqrt[5]))+((((1/2) (1+(Sqrt[5]))))^n) (2+(Sqrt[5])),{n,10}]//FullSimplify

Table[LucasL[n+3],{n,10}]

FullSimplify[-4 Cos[((n π)/2)]-2 Sin[((n π)/2)]/.n->n-1]

seq=Table[(LucasL[n+2]+2 (Cos[((n π)/2)]-2 Sin[((n π)/2)]))/5,{n,20}]

FullSimplify[(LucasL[n+2]+2 (Cos[((n π)/2)]-2 Sin[((n π)/2)]))/5/.n->n+1]

seq=Table[(LucasL[3+n]-2 Sin[((n π)/2)]-4 Cos[((n π)/2)])/5,{n,20}]

FindLinearRecurrence[seq]

LinearRecurrence[{1,0,1,1},{1,3,4,5},20]

FindGeneratingFunction[seq,x]

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

PathComplement

A302689

seq=Table[GraphData[{"PathComplement",n},"TotalDominatingSetCount"],{n,20}]

FindSequenceFunction[Rest@seq,n-1]//FullSimplify

FindGeneratingFunction[seq,x]

Pell

A365571

seq=Table[GraphData[{"Pell",n},"TotalDominatingSetCount"],{n,0,9}]

Prism

Entity["Graph",{"Prism",n}]["TotalDominatingSetCount"]

A296102

seq=Table[GraphData[{"Prism",n},"TotalDominatingSetCount"],{n,3,25}]

seq={3,9,39,121,443,1521,5071,17161,58035,196249,664183,2247001,7601259,25715041,86992799,294294025,995591267,3368061225,11394069191,38545861561,130399710235,441139057489,1492362749807,5048627017225,17079382868243,57779138385081,195465425009943};

FindSequenceFunction[DeleteMissing@seq,n]

coef=FindLinearRecurrence[DeleteMissing@seq]

Length@coef

Take[seq,Length@coef]

LinearRecurrence[{3,0,4,-2,10,4,0,-1,-1},{3,9,39,121,443,1521,5071,17161,58035},20]

gf=x FindGeneratingFunction[DeleteMissing@seq,x]

CoefficientList[Series[((3+12 (x^2)-8 (x^3)+50 (x^4)+24 (x^5)-8 (x^7)-9 (x^8))/(1-3 x-4 (x^3)+2 (x^4)-10 (x^5)-4 (x^6)+(x^8)+(x^9))),{x,0,20}],x]

FullSimplify[SeriesCoefficient[((x (3+12 (x^2)-8 (x^3)+50 (x^4)+24 (x^5)-8 (x^7)-9 (x^8)))/(1-3 x-4 (x^3)+2 (x^4)-10 (x^5)-4 (x^6)+(x^8)+(x^9))),{x,0,n}],n>0]

FullSimplify[%,n∈Integers]

Table[RootSum[-1-#-3 ((#)^2)+((#)^3)&,#^n&]+RootSum[1+#-((#)^2)+((#)^3)&,#^n&]+RootSum[-1+#+((#)^2)+((#)^3)&,#^n&],{n,20}]

Queen n×n

A303207

seq=Table[GraphData[{"Queen",{n,n}},"TotalDominatingSetCount"],{n,20}]

FindSequenceFunction[DeleteMissing@Rest@seq,n-1]//Head//Timing

FindLinearRecurrence[DeleteMissing@Rest@seq]//Head//Timing

FindGeneratingFunction[DeleteMissing@seq,x]//Head//Timing

Rook m×n

Table[GraphData[{"Rook",{m,n}},"TotalDominatingSetCount"],{n,10},{m,n}]//Column//TraditionalForm

Rook n×n

A303208

seq=Table[GraphData[{"Rook",{n,n}},"TotalDominatingSetCount"],{n,20}]

b[0]=1;
b[n_]:=(2^n-1)^n+Sum[Binomial[n,i] Sum[(-1)^j (-1+2^(n-j))^i Binomial[n,j],{j,0,n}],{i,n-1}]

Table[Sum[(((-1))^k)((Binomial[n,k])^2)k!b[n-k],{k,0,n}],{n,20}]

RookComplement n×n

A303209

seq=Table[GraphData[{"RookComplement",{n,n}},"TotalDominatingSetCount"],{n,10}]

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

SierpinskiCarpet

seq=Table[GraphData[{"SierpinskiCarpet",n},"TotalDominatingSetCount"],{n,4}]

SierpinskiSieve

seq=Table[GraphData[{"SierpinskiSieve",n},"TotalDominatingSetCount"],{n,5}]

SierpinskiTetrahedron

seq=Table[GraphData[{"SierpinskiTetrahedron",n},"TotalDominatingSetCount"],{n,6}]

Star

Entity["Graph",{"Star",n}]["TotalDominatingSetCount"]

seq=Table[GraphData[{"Star",n},"TotalDominatingSetCount"],{n,25}]

FindSequenceFunction[DeleteMissing@seq,n]//FullSimplify

Sun

A087131

seq=Table[GraphData[{"Sun",n},"TotalDominatingSetCount"],{n,3,20}]

FindTotalDominatingSet[SunGraph[4],Infinity,All]//Length//Timing

FullSimplify[FindSequenceFunction[DeleteMissing@seq,n-3],n∈Integers]

FindLinearRecurrence[DeleteMissing@seq]

LinearRecurrence[{2,4},{2,12},{0,20}]

seq=Table[2^n LucasL[n],{n,0,20}]

gf=FindGeneratingFunction[DeleteMissing@seq,x]

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

Sunlet

seq=Table[GraphData[{"Sunlet",n},"TotalDominatingSetCount"],{n,3,20}]

FindSequenceFunction[DeleteMissing@seq,n-2]

Tetrahedral

seq=Table[GraphData[{"Tetrahedral",n},"TotalDominatingSetCount"],{n,6,10}]

TorusGrid n×n

A303210

seq=Table[GraphData[{"TorusGrid",{n,n}},"TotalDominatingSetCount"],{n,3,10}]

seq={334,33489,11343461,14511093444,67561889987621,115722034926828228};

Transposition

A303146

Entity["Graph",{"Transposition",n}]["TotalDominatingSetCount"]

seq=Table[GraphData[{"Transposition",n},"TotalDominatingSetCount"],{n,6}]

Sqrt[DeleteMissing[seq]]

FindSequenceFunction[DeleteMissing@seq]

FindGeneratingFunction[DeleteMissing@seq,x]

FindLinearRecurrence[DeleteMissing@Rest@seq]

FindLinearRecurrence[{1,49,12680721}]

bug(350702): FindLinearRecurrence[{1, 49, 12680721}] returns {-1, 12680770} but probably should return unevaluated

Trapezohedron

Entity["Graph",{"Trapezohedron",n}]["TotalDominatingSetCount"]

A362751

seq=Table[GraphData[{"Trapezohedron",n},"TotalDominatingSetCount"],{n,3,20}]

Table[(LucasL[n]+2^n-1)^2,{n,3,20}]==seq

seq=Table[(LucasL[n]+2^n-1)^2,{n,200}];

coef=FindLinearRecurrence[DeleteMissing@seq]

Length@coef

Take[seq,10]

LinearRecurrence[{12,-50,66,84,-270,63,264,-128,-72,32},{4,36,121,484,1764,6561,24336,91204,344569,1313316},200]==seq

FindSequenceFunction[DeleteMissing@seq,n]//Head//Timing

(LucasL[n]+2^n-1)^2/.n->0

FullSimplify[GeneratingFunction[(LucasL[n]+2^n-1)^2,n,x]]

StringReplace[ToString[%,InputForm]," "->""]

SeriesCoefficient[(1/(1-4 x))+(1/(1-x))+(2/(1+x))+(2/(-1+2 x))+((2-3 x)/(1+(-3+x) x))-((2 (-2+x))/(-1+x+(x^2)))+((4 (-1+x))/(-1+2 x+4 (x^2))),{x,0,n}]//FullSimplify

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

FindGeneratingFunction[Take[seq,100],x]//Head//Timing

FindGeneratingFunction[Take[seq,200],x]//Head//Timing

Triangular

A303048

seq=Table[GraphData[{"Triangular",n},"TotalDominatingSetCount"],{n,2,20}]

seq={0,4,54,918,31232,2059624,266734812,68574627036,35160753222400,36021330363615408,73782362964470935112,302225854825997535378632,2475866675779140063716682240,40564755806137338166417907530592,1329227401912999475682655581004557840};

FindSequenceFunction[DeleteMissing@seq]//Head//Timing

FindGeneratingFunction[DeleteMissing@seq,x]//Head//Timing

FindLinearRecurrence[DeleteMissing@Rest@seq]//Head//Timing

b[n_]:=Sum[(((-1))^(n-k))Binomial[n,k](2^(Binomial[k,2])),{k,0,n}]
a[n_]:=Sum[(((-1))^k)Binomial[n,2k](2k)!/((2^k)k!)(b[n-2k]+(n-2k)b[n-2k-1]),{k,0,Floor[n/2]}]

b[n]

b[n_]:=Sum[(((-1))^(n-k))Binomial[n,k](2^(Binomial[k,2])),{k,0,n}]
b[n]

Table[a[n],{n,2,20}]

b[n_]:=Sum[(((-1))^(n-k))Binomial[n,k](2^(Binomial[k,2])),{k,0,n}]
Table[Sum[(((-1))^k)Binomial[n,2k](2k)!/((2^k)k!)(b[n-2k]+(n-2k)b[n-2k-1]),{k,0,Floor[n/2]}],{n,2,20}]

TriangularGrid

A303049

seq=Table[GraphData[{"TriangularGrid",n},"TotalDominatingSetCount"],{n,0,20}]

FindSequenceFunction[DeleteMissing@Rest@seq,n+1]//Head//Timing

FindLinearRecurrence[DeleteMissing@Rest@seq]//Head//Timing

FindGeneratingFunction[DeleteMissing@seq,x]//Head//Timing

TriangularHoneycombAcuteKnight

A304552

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"TotalDominatingSetCount"],{n,20}]

seq={0,0,1,112,3375,64000,3730525,695506456,296474109632,213878375483220,277097671756733625,698661245965852901376,3571973803766628613975275,36637997062083838981741624968};

FindSequenceFunction[DeleteMissing@Drop[seq,2],n-2]//Head//Timing

FindLinearRecurrence[DeleteMissing@Drop[seq,2]]//Head//Timing

FindGeneratingFunction[DeleteMissing@seq,x]//Head//Timing

TriangularHoneycombBishop

A304553

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"TotalDominatingSetCount"],{n,20}]

seq={0,3,29,578,22361,1640001,230504570,62711805029,33349745638405,34939148029874116};

FindSequenceFunction[DeleteMissing@Drop[seq,1],n-1]//Head//Timing

FindLinearRecurrence[DeleteMissing@Drop[seq,1]]//Head//Timing

FindGeneratingFunction[DeleteMissing@seq,x]//Head//Timing

TriangularHoneycombObtuseKnight

A304554

seq=Table[GraphData[{"TriangularHoneycombObtuseKnight",n},"TotalDominatingSetCount"],{n,20}]

seq={0,0,0,0,2464,405185,55169095,11077807074,4422260279456,3897809824077016,7363846127466734748};

FindSequenceFunction[DeleteMissing@Drop[seq,4],n-4]//Head//Timing

FindLinearRecurrence[DeleteMissing@Drop[seq,4]]//Head//Timing

FindGeneratingFunction[DeleteMissing@seq,x]//Head//Timing

TriangularHoneycombQueen

A304556

seq=Table[GraphData[{"TriangularHoneycombQueen",n},"TotalDominatingSetCount"],{n,20}]

seq={0,4,54,928,31338,2060618,266749364,68574754440,35160711883758};

FindSequenceFunction[DeleteMissing@Drop[seq,1],n]//Head//Timing

FindLinearRecurrence[DeleteMissing@Drop[seq,1]]//Head//Timing

FindGeneratingFunction[DeleteMissing@seq,x]//Head//Timing

TriangularHoneycombRook

seq=Table[GraphData[{"TriangularHoneycombRook",n},"TotalDominatingSetCount"],{n,10}]

TriangularSnake

A308592

seq=Table[GraphData[{"TriangularSnake",n},"TotalDominatingSetCount"],{n,1,19,2}]

FindLinearRecurrence[Rest@seq]

seq=LinearRecurrence[{2,3,2},{1,4,16},40]

FullSimplify[FindSequenceFunction[seq,n],n∈Integers]

%19//InputForm

f1=-2-3*#1-2*#1^2+#1^3&;
f2=-81+288*#1-305*#1^2+122*#1^3&;
pos={1,3,2};

(an=ToNumberField[Root[f2,#],Root[f1,pos[[#]]]]&/@Range[3])//InputForm

coeffs=an[[All,-1]]

SameQ@@coeffs

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

Table[RootSum[-2-3 #-2 ((#)^2)+((#)^3)&,65 ((#)^n)-100 ((#)^(n+1))+31 ((#)^(n+2))&]/122,{n,20}]

gf=x FindGeneratingFunction[seq,x]

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

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

Web

seq=Table[GraphData[{"Web",n},"TotalDominatingSetCount"],{n,3,20}]

FindSequenceFunction[DeleteMissing@seq,n-2]

Wheel

A302603

Entity["Graph",{"Wheel",n}]["TotalDominatingSetCount"]

seq=Table[GraphData[{"Wheel",n},"TotalDominatingSetCount"],{n,4,20}]

FullSimplify[FindSequenceFunction[DeleteMissing@seq,n-3],n∈Integers]

gf=(x^4)FindGeneratingFunction[DeleteMissing@seq,x]//FullSimplify

FullSimplify[SeriesCoefficient[gf,{x,0,n}],n∈Integers&&n>3]

Table[(2^(-1-n)) (-(((1-(Sqrt[5])))^n) (1+(Sqrt[5]))+(-1+(Sqrt[5])) (((1+(Sqrt[5])))^n)+(2^n) (-2+Cosh[n Log[2]]+4 Sin[((n π)/2)]+Sinh[n Log[2]])),{n,20}]//FullSimplify

Table[(2^(-1-n)) ((2^(n+1))LucasL[n-1]+(2^n) (-2+Cosh[n Log[2]]+4 Sin[((n π)/2)]+Sinh[n Log[2]])),{n,20}]//FullSimplify

FullSimplify[(2^(-1-n)) ((2^(n+1))LucasL[n-1]+(2^n) (-2+Cosh[n Log[2]]+4 Sin[((n π)/2)]+Sinh[n Log[2]]))]

Table[LucasL[n-1]+2 Sin[((n π)/2)]-1+(Cosh[n Log[2]]+Sinh[n Log[2]])/2,{n,20}]//FullSimplify

FullSimplify[(Cosh[n Log[2]]+Sinh[n Log[2]])/2]

Table[(Cosh[n Log[2]]+Sinh[n Log[2]])/2,{n,5}]//FullSimplify

seq=Table[LucasL[n-1]+2 Sin[((n π)/2)]+(2^(n-1))-1,{n,50}]

gf=x FindGeneratingFunction[seq,x]

InputForm[gf]

CoefficientList[Series[((-4+14 x-16 (x^2)+7 (x^3)-2 (x^4)+3 (x^5))/(-1+4 x-5 (x^2)+3 (x^3)-2 (x^4)-(x^5)+2 (x^6))),{x,0,20}],x]

FindLinearRecurrence[seq]

LinearRecurrence[{4,-5,3,-2,-1,2},{4,2,4,11,24,42},20]

WhiteBishop n×n

A303147

seq=Table[GraphData[{"WhiteBishop",{n,n}},"TotalDominatingSetCount"],{n,2,10}]