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

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

Author

Eric W. Weisstein, Apr 11 2018

Keywords

• nonn

Comments

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

Links

• 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).

Crossrefs

Cf. A213661, A290270, A300738, A302603.

Programs

Mathematica

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]

PARI

{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

Formula

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)).

Extensions

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