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A291703
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Number of connected dominating sets in the complete tripartite graph K_{n,n,n}.
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2
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7, 54, 490, 4050, 32674, 261954, 2096770, 16776450, 134216194, 1073738754, 8589928450, 68719464450, 549755789314, 4398046461954, 35184371990530, 281474976514050, 2251799813292034, 18014398508695554, 144115188074283010, 1152921504603701250
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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FORMULA
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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)
E.g.f.: 3*x + exp(x)*(2 - 3*exp(x) + exp(7*x)). - Stefano Spezia, Dec 24 2021
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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