OFFSET
1,1
COMMENTS
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.
a(n) = Sum_{k>=1} A213654(n,k).
REFERENCES
S. Alikhani and Y. H. Peng, Dominating sets and domination polynomials of certain graphs, II, Opuscula Math., 30, No. 1, 2010, 37-51.
LINKS
S. Alikhani and Y. H. Peng, Introduction to domination polynomial of a graph, arXiv:0905.2251.
Index entries for linear recurrences with constant coefficients, signature (1,1,1).
FORMULA
a(n) = a(n-1) + a(n-2) + a(n-3) for n >= 4; a(1)=13, a(2)=23, a(3)=41.
G.f.: -x*(13+10*x+5*x^2)/(-1+x+x^2+x^3). - R. J. Mathar, Jul 22 2022
EXAMPLE
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}.
MAPLE
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);
MATHEMATICA
LinearRecurrence[{1, 1, 1}, {13, 23, 41}, 30] (* Jean-François Alcover, Dec 02 2017 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Emeric Deutsch, Jun 18 2012
STATUS
approved