OFFSET
1,1
COMMENTS
For n > 1, the only dominating sets that are not connected are the 3 sets consisting of all vertices from a single partition. For n > 1, the only connected sets that are not dominating are the 3n sets consisting of a single vertex. - Andrew Howroyd, Aug 30 2017
LINKS
Colin Barker, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Complete Tripartite Graph
Eric Weisstein's World of Mathematics, Connected Dominating Set
Index entries for linear recurrences with constant coefficients, signature (11,-26,16).
FORMULA
From Colin Barker, Sep 01 2017: (Start)
G.f.: x*(7 - 23*x + 78*x^2 - 48*x^3) / ((1 - x)*(1 - 2*x)*(1 - 8*x)).
a(n) = 2 - 3*2^n + 8^n for n > 1.
a(n) = 11*a(n-1) - 26*a(n-2) + 16*a(n-3) for n > 4.
(End)
a(n) = A302942(n) for n > 1. - Eric W. Weisstein, Apr 16 2018
E.g.f.: 3*x + exp(x)*(2 - 3*exp(x) + exp(7*x)). - Stefano Spezia, Dec 24 2021
MATHEMATICA
a[1] = 7; a[n_] := 2 - 3*2^n + 8^n; Array[a, 20] (* Jean-François Alcover, Oct 08 2017 *)
(* Start from Eric W. Weisstein, Oct 12 2017 *)
Table[Piecewise[{{7, n == 1}}, 2 - 3 2^n + 8^n], {n, 20}]
Table[Piecewise[{{7, n == 1}}, 2 (2^(n - 1) + 1) (2^n - 1)^2], {n, 20}]
Join[{7}, LinearRecurrence[{11, -26, 16}, {54, 490, 4050}, 20]]
CoefficientList[Series[(-7 + 23 x - 78 x^2 + 48 x^3)/(-1 + 11 x - 26 x^2 + 16 x^3), {x, 0, 20}], x]
(* End *)
PROG
(PARI) a(n) = if(n < 2, [7][n], 8^n - 3*2^n + 2); \\ Andrew Howroyd, Aug 30 2017
(PARI) Vec(x*(7 - 23*x + 78*x^2 - 48*x^3) / ((1 - x)*(1 - 2*x)*(1 - 8*x)) + O(x^30)) \\ Colin Barker, Sep 01 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Eric W. Weisstein, Aug 30 2017
EXTENSIONS
a(8)-a(20) from Andrew Howroyd, Aug 30 2017
STATUS
approved