A291537 -id:A291537 - cp's OEIS frontend

A291703 Number of connected dominating sets in the complete tripartite graph K_{n,n,n}.

7, 54, 490, 4050, 32674, 261954, 2096770, 16776450, 134216194, 1073738754, 8589928450, 68719464450, 549755789314, 4398046461954, 35184371990530, 281474976514050, 2251799813292034, 18014398508695554, 144115188074283010, 1152921504603701250

Offset: 1

Eric W. Weisstein, Aug 30 2017

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

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).

Cf. A290756, A291537.

Cf. A302942.

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 *)

a(n) = if(n < 2, [7][n], 8^n - 3*2^n + 2); \\ Andrew Howroyd, Aug 30 2017

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

a(n) = A291537(n) - 3 = A290756(n) - 3*n for n > 1. - Andrew Howroyd, Aug 30 2017
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

a(8)-a(20) from Andrew Howroyd, Aug 30 2017

A347536 Number of minimum dominating sets in the complete tripartite graph K_{n,n,n}.

Original entry on oeis.org

3, 15, 27, 48, 75, 108, 147, 192, 243, 300, 363, 432, 507, 588, 675, 768, 867, 972, 1083, 1200, 1323, 1452, 1587, 1728, 1875, 2028, 2187, 2352, 2523, 2700, 2883, 3072, 3267, 3468, 3675, 3888, 4107, 4332, 4563, 4800, 5043, 5292, 5547, 5808, 6075, 6348, 6627

Offset: 1

Eric W. Weisstein, Sep 05 2021

Eric Weisstein's World of Mathematics, Complete Tripartite Graph
Eric Weisstein's World of Mathematics, Minimum Dominating Set
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A291537, A290325.

a(n) = A033428(n) = 3*n^2 for n != 2.
G.f.: 3*x*(-1 - 2*x + 3*x^2 - 3*x^3 + x^4)/(-1 + x)^3.
From Stefano Spezia, Sep 06 2021: (Start)
E.g.f.: 3*x*(x + 2*exp(x)*(1 + x))/2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 5. (End)

cp's OEIS Frontend

A291703 Number of connected dominating sets in the complete tripartite graph K_{n,n,n}.

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