1,1

Number of dominating subsets of the graph G(n) consisting of an edge ab and n triangles, each having one vertex identified with the vertex b.

Resultant of polynomial 2*x^n+1 with the polynomial 3*x^(n+1)+2. - Philippe Deléham, Jan 19 2024

Vincenzo Librandi, Table of n, a(n) for n = 1..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 (7,-12).

a(n) = Sum_{k=1..2*n+2} A213658(n,k).

a(n) = 3^n + 2^(2*n+1).

G.f.: -x*(-11+36*x) / ( (4*x-1)*(3*x-1) ). - R. J. Mathar, Jul 03 2012

a(n) = 7*a(n-1) - 12*a(n-2). - Philippe Deléham, Jan 19 2024

a(1)=11 because the graph G(1) is abcd with edges ab, bc, bd, and cd; there is 1 dominating subset of size 1 ({b}), all binomial(4,2)=6 subsets of size 2 of {a,b,c,d} with the exception of {c,d} are dominating; all binomial(4,3)=4 subsets of size 3 of {a,b,c,d} are dominating; obviously, {a,b,c,d} is dominating; 1+5+4+1 = 11.

a(1) = Det [2, 1, 0; 0, 2, 1; 3, 0, 2] = 11; a(2) = Det [2, 0, 1, 0, 0; 0, 2, 0, 1, 0; 0, 0, 2, 0, 1; 3, 0, 0, 2, 0; 0, 3, 0, 0, 2] = 41; a(3) = Det [2, 0, 0, 1, 0, 0, 0; 0, 2, 0, 0, 1, 0, 0; 0, 0, 2, 0, 0, 1, 0; 0, 0, 0, 2, 0, 0, 1; 3, 0, 0, 0, 2, 0, 0; 0, 3, 0, 0, 0, 2, 0; 0, 0, 3, 0, 0, 0, 2] = 155. - Philippe Deléham, Jan 19 2024

seq(3^n+2^(2*n+1), n=1..30);

Table[3^n + 2^(2 n + 1), {n, 40}] (* Vincenzo Librandi, Jun 30 2017 *)

LinearRecurrence[{7, -12}, {11, 41}, 30] (* Harvey P. Dale, Sep 10 2017 *)

(Magma) [3^n+2^(2*n+1): n in [1..30]]; // Vincenzo Librandi, Jun 30 2017

Cf. A000244, A004171, A213658.

Sequence in context: A139933 A243892 A288795 * A199209 A156733 A079304

Adjacent sequences: A213656 A213657 A213658 * A213660 A213661 A213662

nonn,easy

Emeric Deutsch, Jun 29 2012

approved