OFFSET
1,1
LINKS
S. Alikhani and Emeric Deutsch, More on domination polynomial and domination root (Previous title: "Graphs with domination roots in the right half-plane"), arXiv preprint arXiv:1305.3734 [math.CO], 2013-2014.
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 (3,-1,-1,-2)
FORMULA
a(n) = Sum(k=1..n+2) A213662(n,k).
a(n) = a(n-1) + a(n-2) + a(n-3) + 2^(n-2) for n>=4. [corrected by Harvey P. Dale, Aug 11 2012]
From R. J. Mathar, Jul 03 2012: (Start)
a(n) = 2^(n+1) + A000213(n+2).
G.f.: -x*(-7+8*x+7*x^2+6*x^3) / ( (2*x-1)*(x^3+x^2+x-1) ). (End)
EXAMPLE
a(1)=7 because the graph G(1) is the triangle abc; there are 3 dominating subsets of size 1 ({a}, {b}, {c}), 3 dominating subsets of size 2 ({a,b}, {a,c}, {b,c}), and 1 dominating subset of size 3 ({a,b,c}).
MAPLE
a[1] := 7: a[2] := 13: a[3] := 25: for n from 4 to 35 do a[n] := a[n-1]+a[n-2]+a[n-3]+2^(n-2) end do: seq(a[n], n = 1 .. 35);
MATHEMATICA
RecurrenceTable[{a[1]==7, a[2]==13, a[3]==25, a[n]==a[n-1]+a[n-2]+ a[n-3]+ 2^(n-2)}, a, {n, 40}] (* Harvey P. Dale, Aug 11 2012 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Jun 29 2012
STATUS
approved