A290325 Number of minimal dominating sets (and maximal irredundant sets) in the complete tripartite graph K_{n,n,n}.

3, 15, 30, 51, 78, 111, 150, 195, 246, 303, 366, 435, 510, 591, 678, 771, 870, 975, 1086, 1203, 1326, 1455, 1590, 1731, 1878, 2031, 2190, 2355, 2526, 2703, 2886, 3075, 3270, 3471, 3678, 3891, 4110, 4335, 4566, 4803, 5046, 5295, 5550, 5811, 6078

Offset

1,1

Comments

When n>1 the minimal dominating sets consist of either a single vertex from any two of the partitions or all vertices from just one of the partitions. When n=1 only the later are minimal. - Andrew Howroyd, Jul 27 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, Maximal Irredundant Set

Eric Weisstein's World of Mathematics, Minimal Dominating Set

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Formula

a(n) = 3*n^2 + 3 for n > 1. - Andrew Howroyd, Jul 27 2017

From Colin Barker, Jul 27 2017: (Start)

G.f.: 3*x*(1 + 2*x - 2*x^2 + x^3) / (1 - x)^3.

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 4. (End)

E.g.f.: 3((x^2 + x + 1)*exp(x) - (2*x + 1)) + 3*x. - G. C. Greubel, Aug 17 2017

Mathematica

Rest[With[{nn = 50}, CoefficientList[Series[3 ((x^2 + x + 1)*Exp[x] - (2*x + 1)) + 3*x, {x, 0, nn}], x]*Range[0, nn]!]] (* or *) Table[3*(n^2 +1), {n, 1, 50}] (* G. C. Greubel, Aug 17 2017 *)

Prog

(PARI) Vec(3*x*(1 + 2*x - 2*x^2 + x^3) / (1 - x)^3 + O(x^60)) \\ Colin Barker, Jul 27 2017

Crossrefs

Sequence in context: A087183 A297851 A298088 * A271326 A106354 A228308

Adjacent sequences: A290322 A290323 A290324 * A290326 A290327 A290328

Keyword

nonn,easy

Author

Eric W. Weisstein, Jul 27 2017

Extensions

a(6)-a(45) from Andrew Howroyd, Jul 27 2017

Maximal irredundant sets added to name by Eric W. Weisstein, Aug 17 2017

Status

approved