A213665 Number of dominating subsets of the graph G(n) obtained by joining a vertex with two consecutive vertices of the cycle graph C_n (n >= 3).

13, 23, 43, 79, 145, 267, 491, 903, 1661, 3055, 5619, 10335, 19009, 34963, 64307, 118279, 217549, 400135, 735963, 1353647, 2489745, 4579355, 8422747, 15491847, 28493949, 52408543, 96394339, 177296831, 326099713, 599790883, 1103187427

Offset

3,1

Links

Vincenzo Librandi, Table of n, a(n) for n = 3..1000

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.

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

Formula

a(n) = Sum_{k=1..n+1} A213664(n,k).

a(n) = a(n-1) + a(n-2) + a(n-3) for n >= 6.

G.f.: x^3 * (13+10*x+7*x^2) / (1-x-x^2-x^3). - R. J. Mathar, Jul 03 2012

Example

a(3)=13 because G(3) is the square abcd with the additional edge bd; all nonempty subsets of {a,b,c,d} are dominating, with the exception of {a} and {c}: 2^4 - 1 - 2 = 13.

Maple

a[3] := 13: a[4] := 23: a[5] := 43: for n from 6 to 42 do a[n] := a[n-1]+a[n-2]+a[n-3] end do: seq(a[n], n = 3 .. 42);

Mathematica

CoefficientList[Series[(13+10*x+7*x^2)/(1-x-x^2-x^3), {x, 0, 100}], x] (* Vincenzo Librandi, Aug 03 2012 *)
LinearRecurrence[{1, 1, 1}, {13, 23, 43}, 40] (* Harvey P. Dale, Dec 11 2012 *)

Crossrefs

Cf. A213664.

Sequence in context: A240113 A256177 A320752 * A068712 A103166 A154863

Adjacent sequences: A213662 A213663 A213664 * A213666 A213667 A213668

Keyword

nonn,easy

Author

Emeric Deutsch, Jun 30 2012

Status

approved