Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #17 Jan 14 2024 20:33:42
%S 13,23,41,77,141,259,477,877,1613,2967,5457,10037,18461,33955,62453,
%T 114869,211277,388599,714745,1314621,2417965,4447331,8179917,15045213,
%U 27672461,50897591,93615265,172185317,316698173,582498755
%N Number of dominating subsets of the theta-graph TH(2,2,n) (n>=1). A tribonacci sequence with initial values 13, 23, and 41.
%C A theta-graph is a graph consisting of two vertices of degree three, connected by three paths of one or more edges each. In the theta-graph TH(2,2,n) the three paths have 2, 2, and n edges, respectively.
%C a(n) = Sum_{k>=1} A213654(n,k).
%D S. Alikhani and Y. H. Peng, Dominating sets and domination polynomials of certain graphs, II, Opuscula Math., 30, No. 1, 2010, 37-51.
%H S. Alikhani and Y. H. Peng, <a href="http://arxiv.org/abs/0905.2251"> Introduction to domination polynomial of a graph</a>, arXiv:0905.2251.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,1).
%F a(n) = a(n-1) + a(n-2) + a(n-3) for n >= 4; a(1)=13, a(2)=23, a(3)=41.
%F G.f.: -x*(13+10*x+5*x^2)/(-1+x+x^2+x^3). - _R. J. Mathar_, Jul 22 2022
%e a(1)=13. TH(2,2,1) is the graph obtained from the cycle ABCD by joining vertices A and C. All 2^4 - 1 = 15 nonempty subsets of {A,B,C,D} are dominating with the exception of {B} and {D}.
%p a := proc (n) if n = 1 then 13 elif n = 2 then 23 elif n = 3 then 41 else a(n-1)+a(n-2)+a(n-3) end if end proc: seq(a(n), n = 1 .. 30);
%t LinearRecurrence[{1, 1, 1}, {13, 23, 41}, 30] (* _Jean-François Alcover_, Dec 02 2017 *)
%Y Cf. A213654.
%K nonn,easy
%O 1,1
%A _Emeric Deutsch_, Jun 18 2012