A300738 -id:A300738 - cp's OEIS frontend

A302655 Number of minimal total dominating sets in the n-path graph.

0, 1, 2, 1, 2, 4, 3, 4, 8, 9, 10, 16, 21, 25, 36, 49, 60, 81, 112, 144, 189, 256, 336, 441, 592, 784, 1029, 1369, 1820, 2401, 3182, 4225, 5586, 7396, 9815, 12996, 17200, 22801, 30210, 40000, 53001, 70225, 93000, 123201, 163240, 216225, 286416, 379456, 502665

Offset: 1

Eric W. Weisstein, Apr 11 2018

Andrew Howroyd, Table of n, a(n) for n = 1..200
Eric Weisstein's World of Mathematics, Minimal Total Dominating Set.
Eric Weisstein's World of Mathematics, Path Graph.
Index entries for linear recurrences with constant coefficients, signature (0,0,1,1,1,1,0,-1,-1).

Row 1 of A303118.

Cf. A000931, A300738, A302654.

Table[If[Mod[n, 2] == 0, (RootSum[-1 - # + #^3 &, #^(n/2 + 5) (5 - 6 # + 4 #^2) &]/23)^2, (RootSum[-1 + # - 2 #^2 + #^3 &, #^((n - 1)/2) (4 - 2 # + 5 #^2) &] + RootSum[-1 + #^2 + #^3 &, #^((n - 1)/2) (-5 + 6 # + 3 #^2) &])/23], {n, 50}]
LinearRecurrence[{0, 0, 1, 1, 1, 1, 0, -1, -1}, {0, 1, 2, 1, 2, 4, 3, 4, 8}, 50]
CoefficientList[Series[(x (1 + 2 x + x^2 + x^3 + x^4 - x^5 - 2 x^6 - x^7))/(1 - x^3 - x^4 - x^5 - x^6 + x^8 + x^9), {x, 0, 50}], x]

concat([0],Vec(x^2*(1 + 2*x + x^2 + x^3 + x^4 - x^5 - 2*x^6 - x^7)/(1 - x^3 - x^4 - x^5 - x^6 + x^8 + x^9) + O(x^50))) \\ Andrew Howroyd, Apr 15 2018

From Andrew Howroyd, Apr 15 2018: (Start)
a(n) = a(n-3) + a(n-4) + a(n-5) + a(n-6) - a(n-8) - a(n-9) for n > 9.
G.f.: x^2*(1 + 2*x + x^2 + x^3 + x^4 - x^5 - 2*x^6 - x^7)/(1 - x^3 - x^4 - x^5 - x^6 + x^8 + x^9).
a(2*n) = A000931(n+5)^2. (End)

Terms a(20) and beyond from Andrew Howroyd, Apr 15 2018

A302918 Number of nonequivalent minimal total dominating sets in the n-cycle graph up to rotation.

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 1, 1, 2, 3, 2, 4, 3, 4, 6, 7, 7, 10, 11, 17, 19, 23, 28, 38, 46, 60, 75, 96, 120, 160, 197, 257, 327, 420, 539, 701, 892, 1155, 1488, 1928, 2479, 3220, 4148, 5381, 6961, 9030, 11687, 15183, 19673, 25563, 33174, 43128, 56010, 72864, 94719

Offset: 1

Andrew Howroyd, Apr 15 2018

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Cycle Graph.
Eric Weisstein's World of Mathematics, Minimal Total Dominating Set.

Cf. A300738.

A300738 = DifferenceRoot[Function[{f, n}, {f[n] + f[n+1] - f[n+3] - f[n+4] - f[n+5] - f[n+6] + f[n+9] == 0, f[1]==0, f[2]==0, f[3]==3, f[4]==4, f[5]==5, f[6]==9, f[7]==7, f[8]==4, f[9]==12}]];
a[n_] := (1/n) Sum[EulerPhi[n/d] A300738[d], {d, Divisors[n]}];
a /@ Range[1, 55] (* Jean-François Alcover, Sep 21 2019 *)

NecklaceT(v)={vector(#v, n, sumdiv(n,d,eulerphi(n/d)*v[d])/n)}
NecklaceT(concat([0,0],Vec((3 + 4*x + 5*x^2 + 6*x^3 - 8*x^5 - 9*x^6)/((1 - x^2 - x^3)*(1 + x^2 - x^6)) + O(x^50))))

a(n) = (1/n) * Sum_{d|n} phi(n/d) * A300738(d).

A362812 Number of minimal total dominating sets in the n-double cone graph.

Original entry on oeis.org

15, 24, 35, 93, 63, 32, 162, 645, 506, 649, 1547, 2429, 4654, 10032, 14195, 20772, 43719, 83561, 134731, 234300, 414782, 707329, 1276950, 2313493, 3932343, 6765257, 12110458, 21381436, 37295511, 65610064, 114854155, 200533989, 353703319, 623201368

Offset: 3

Eric W. Weisstein, May 04 2023

nonn

Andrew Howroyd, Table of n, a(n) for n = 3..1000
Eric Weisstein's World of Mathematics, Double Cone Graph
Eric Weisstein's World of Mathematics, Minimal Total Dominating Set

Cf. A300738, A362750.

a(n) = 2*n + A300738(n)^2. - Andrew Howroyd, May 04 2023

Terms a(13) and beyond from Andrew Howroyd, May 04 2023

A302658 Number of minimal total dominating sets in the wheel graph on n nodes.

Original entry on oeis.org

1, 2, 6, 8, 10, 15, 14, 12, 21, 35, 33, 37, 52, 63, 83, 116, 136, 162, 228, 309, 388, 506, 667, 865, 1155, 1547, 2010, 2629, 3509, 4654, 6138, 8132, 10750, 14195, 18842, 25000, 33041, 43719, 57957, 76769, 101680, 134731, 178407, 236240, 313052, 414782, 549336

Offset: 2

Eric W. Weisstein, Apr 11 2018

nonn

Wheel graphs are defined for n>=4; extended to n=2 using formula. - Andrew Howroyd, Apr 15 2018

Eric Weisstein's World of Mathematics, Minimal Total Dominating Set.
Eric Weisstein's World of Mathematics, Wheel Graph.
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-1,0,0,-1,0,1,1,-1).

Cf. A213661, A290270, A300738, A302603.

Table[n - 1 + RootSum[-1 - # + #^3 &, #^(n - 1) &] + (1 - (-1)^n) RootSum[-1 + #^2 + #^3 &, #^((n - 1)/2) &], {n, 2, 50}]
LinearRecurrence[{2, -1, 1, -1, 0, 0, -1, 0, 1, 1, -1}, {1, 2, 6, 8, 10, 15, 14, 12, 21, 35, 33}, 50]
CoefficientList[Series[(1 + 3 x^2 - 3 x^3 - x^4 - x^5 - 8 x^6 - 2 x^7 + 8 x^8 + 11 x^9 - 9 x^10)/((-1 + x)^2 (1 - x^3 - x^4 - x^5 - x^6 + x^8 + x^9)), {x, 0, 50}], x]

{my(v=concat([0,0],Vec((3 + 4*x + 5*x^2 + 6*x^3 - 8*x^5 - 9*x^6)/((1 - x^2 - x^3)*(1 + x^2 - x^6)) + O(x^50))));vector(#v,i,v[i]+i)} \\ Andrew Howroyd, Apr 15 2018

a(n) = A300738(n-1) + (n-1). - Andrew Howroyd, Apr 15 2018

G.f.: x^2*(1 + 3*x^2 - 3*x^3 - x^4 - x^5 - 8*x^6 - 2*x^7 + 8*x^8 + 11*x^9 - 9*x^10)/((-1 + x)^2*(1 - x^3 - x^4 - x^5 - x^6 + x^8 + x^9)).

a(2)-a(3) and terms a(20) and beyond from Andrew Howroyd, Apr 15 2018

cp's OEIS Frontend

A302655 Number of minimal total dominating sets in the n-path graph.

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A302918 Number of nonequivalent minimal total dominating sets in the n-cycle graph up to rotation.

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A362812 Number of minimal total dominating sets in the n-double cone graph.

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A302658 Number of minimal total dominating sets in the wheel graph on n nodes.

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