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A292074
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Number of minimum dominating sets in the n X n rook complement graph.
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3
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1, 4, 48, 240, 1000, 3300, 9114, 21952, 47520, 94500, 175450, 307824, 515112, 828100, 1286250, 1939200, 2848384, 4088772, 5750730, 7942000, 10789800, 14443044, 19074682, 24884160, 32100000, 40982500, 51826554, 64964592, 80769640, 99658500, 122095050
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OFFSET
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1,2
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COMMENTS
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The minimum dominating sets are the minimal dominating sets (A291623) of size equal to the domination number. For n > 2, the domination number is 3. For n > 3, the minimal dominating sets of size 3 are either any three vertices such that no two are in the same row or column or any vertex with another in the same row and a third in the same column. For n = 3, the case of all vertices in a single row or column also needs to be included. - Andrew Howroyd, Sep 13 2017
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LINKS
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FORMULA
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G.f.: x*(-1 + 3*x - 41*x^2 + 47*x^3 - 223*x^4 + 221*x^5 - 217*x^6 + 127*x^7 - 42*x^8 + 6*x^9)/(-1 + x)^7.
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n > 10.
a(n) = 6*binomial(n, 3)^2 + n^2*(n-1)^2 for n > 3. - Andrew Howroyd, Sep 13 2017
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MATHEMATICA
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Join[{1, 4, 48}, LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {240, 1000, 3300, 9114, 21952, 47520, 94500}, 20]]
Table[Piecewise[{{1, n == 1}, {48, n == 3}}, 6 Binomial[n, 3]^2 + n^2 (n - 1)^2], {n, 20}]
CoefficientList[Series[(-1 + 3 x - 41 x^2 + 47 x^3 - 223 x^4 + 221 x^5 - 217 x^6 + 127 x^7 - 42 x^8 + 6 x^9)/(-1 + x)^7, {x, 0, 20}], x]
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PROG
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(PARI) a(n) = if(n<4, [1, 4, 48][n], 6*binomial(n, 3)^2 + n^2*(n-1)^2); \\ Andrew Howroyd, Sep 13 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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