A213661 Number of dominating subsets of the wheel graph W_n.

4, 3, 7, 15, 27, 53, 103, 199, 387, 753, 1467, 2863, 5595, 10949, 21455, 42095, 82691, 162625, 320179, 631031, 1244907, 2458261, 4858487, 9610231, 19024131, 37687153, 74710123, 148198623, 294150331, 584167941, 1160734623, 2307488351, 4589261827

Offset

1,1

Comments

a(n) = Sum_{k=1..n} A212635(n,k).

Extended to a(1)-a(3) using the formula/recurrence.

Links

Table of n, a(n) for n=1..33.

S. Alikhani and Y. H. Peng, Introduction to domination polynomial of a graph, arXiv:0905.2251 [math.CO], 2009.

T. Kotek, J. Preen, F. Simon, P. Tittmann, and M. Trinks, Recurrence relations and splitting formulas for the domination polynomial, arXiv:1206.5926 [math.CO], 2012.

Eric Weisstein's World of Mathematics, Dominating Set

Eric Weisstein's World of Mathematics, Wheel Graph

Index entries for linear recurrences with constant coefficients, signature (3,-1,-1,-2).

Formula

a(n) = a(n-1) + a(n-2) + a(n-3) + 2^(n-4) for n >= 4.

G.f.: x*(4 - 9*x + 2*x^2 + x^3)/(1 - 3*x + x^2 + x^3 + 2*x^4).

a(n) = 2^(n-1) -A000073(n+2)+4*A000073(n+1) -A000073(n). - R. J. Mathar, Jun 29 2012

a(n) = 3*a(n-1) - a(n-2) - a(n-3) - 2*a(n-4). - Eric W. Weisstein, Apr 17 2018

Example

a(4)=15 because all nonempty subsets of the wheel W_4 are dominating (2^4 - 1 = 15).

Maple

a[4] := 15: a[5] := 27: a[6] := 53: for n from 7 to 42 do a[n] := a[n-1]+a[n-2]+a[n-3]+2^(n-4) end do: seq(a[n], n = 4 .. 40);

Mathematica

LinearRecurrence[{3, -1, -1, -2}, {4, 3, 7, 15}, 40] (* Eric W. Weisstein, Mar 31 2017 *)
Table[2^(n - 1) + RootSum[-1 - # - #^2 + #^3 &, #^n (-1 - # + #1^2) &], {n, 20}] (* Eric W. Weisstein, Apr 17 2018 *)
CoefficientList[Series[(4 - 9 x + 2 x^2 + x^3)/(1 - 3 x + x^2 + x^3 + 2 x^4), {x, 0, 20}], x] (* Eric W. Weisstein, Apr 17 2018 *)

Crossrefs

Cf. A212635.

Sequence in context: A245300 A228949 A048227 * A176083 A092193 A277117

Adjacent sequences: A213658 A213659 A213660 * A213662 A213663 A213664

Keyword

nonn,easy

Author

Emeric Deutsch, Jun 29 2012

Extensions

a(1)-a(3) prepended by Eric W. Weisstein, Apr 17 2018

Status

approved