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 A291623 Number of maximal irredundant and minimal dominating sets in the n X n rook complement graph. 4
 1, 4, 48, 320, 1610, 6012, 17948, 45488, 101970, 207920, 393272, 699888, 1184378, 1921220, 3006180, 4560032, 6732578, 9706968, 13704320, 18988640, 25872042, 34720268, 45958508, 60077520, 77640050, 99287552, 125747208, 157839248, 196484570, 242712660 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS From Andrew Howroyd, Aug 30 2017: (Start) For n > 2 the maximal irredundant sets are: - all vertices in any single row or column, - any three vertices such that no two are in the same row or column, - any vertex with another in the same row and a third in the same column, - two vertices in each of two rows/columns and none in the same column/row. (End) LINKS Andrew Howroyd, Table of n, a(n) for n = 1..500 Eric Weisstein's World of Mathematics, Maximal Irredundant Set Eric Weisstein's World of Mathematics, Minimal Dominating Set Eric Weisstein's World of Mathematics, Rook Complement Graph Index entries for linear recurrences with constant coefficients, signature (7, -21, 35, -35, 21, -7, 1). FORMULA From Andrew Howroyd, Aug 30 2017: (Start) a(n) = 2*n + 6*binomial(n,3)^2 + n^2*(n-1)^2 + 12*binomial(n,4)*binomial(n,2) for n > 2. a(n) = (5*n^6 - 33*n^5 + 89*n^4 - 99*n^3 + 38*n^2 + 24*n) / 12 for n > 2. 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 > 9. G.f.: x*(1 - 3*x + 41*x^2 + 33*x^3 + 273*x^4 - 99*x^5 + 77*x^6 - 27*x^7 + 4*x^8)/(1-x)^7. (End) MATHEMATICA Table[Piecewise[{{1, n == 1}, {4, n == 2}}, 2 n + 6 Binomial[n, 3]^2 + n^2 (n - 1)^2 + 12 Binomial[n, 4] Binomial[n, 2]], {n, 20}] Join[{1, 4}, LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {48, 320, 1610, 6012, 17948, 45488, 101970}, 18]] CoefficientList[Series[(-1 + 3 x - 41 x^2 - 33 x^3 - 273 x^4 + 99 x^5 - 77 x^6 + 27 x^7 - 4 x^8)/(-1 + x)^7, {x, 0, 20}], x] PROG (PARI) a(n) = if(n<3, [1, 4][n], (5*n^6 - 33*n^5 + 89*n^4 - 99*n^3 + 38*n^2 + 24*n) / 12); \\ Andrew Howroyd, Aug 30 2017 CROSSREFS Cf. A291622. Sequence in context: A274729 A217153 A129002 * A144704 A091904 A192831 Adjacent sequences: A291620 A291621 A291622 * A291624 A291625 A291626 KEYWORD nonn AUTHOR Eric W. Weisstein, Aug 28 2017 EXTENSIONS Terms a(6) and beyond from Andrew Howroyd, Aug 30 2017 STATUS approved

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Last modified September 16 22:04 EDT 2024. Contains 375979 sequences. (Running on oeis4.)