A290325 - OEIS

%I #22 Aug 17 2017 22:06:25

%S 3,15,30,51,78,111,150,195,246,303,366,435,510,591,678,771,870,975,

%T 1086,1203,1326,1455,1590,1731,1878,2031,2190,2355,2526,2703,2886,

%U 3075,3270,3471,3678,3891,4110,4335,4566,4803,5046,5295,5550,5811,6078

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

%C 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

%H Colin Barker, <a href="/A290325/b290325.txt">Table of n, a(n) for n = 1..1000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CompleteTripartiteGraph.html">Complete Tripartite Graph</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MaximalIrredundantSet.html">Maximal Irredundant Set</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MinimalDominatingSet.html">Minimal Dominating Set</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F a(n) = 3*n^2 + 3 for n > 1. - _Andrew Howroyd_, Jul 27 2017

%F From _Colin Barker_, Jul 27 2017: (Start)

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

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

%F E.g.f.: 3((x^2 + x + 1)*exp(x) - (2*x + 1)) + 3*x. - _G. C. Greubel_, Aug 17 2017

%t 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 *)

%o (PARI) Vec(3*x*(1 + 2*x - 2*x^2 + x^3) / (1 - x)^3 + O(x^60)) \\ _Colin Barker_, Jul 27 2017

%K nonn,easy

%O 1,1

%A _Eric W. Weisstein_, Jul 27 2017

%E a(6)-a(45) from _Andrew Howroyd_, Jul 27 2017

%E Maximal irredundant sets added to name by _Eric W. Weisstein_, Aug 17 2017