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A039623
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a(n) = n^2*(n^2+3)/4.
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14
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1, 7, 27, 76, 175, 351, 637, 1072, 1701, 2575, 3751, 5292, 7267, 9751, 12825, 16576, 21097, 26487, 32851, 40300, 48951, 58927, 70357, 83376, 98125, 114751, 133407, 154252, 177451, 203175, 231601, 262912, 297297, 334951, 376075, 420876, 469567
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OFFSET
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1,2
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COMMENTS
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Previous definition was: Consider a figure like this <> (a squashed square, symmetric about both axes); each side is given 1 of n colors; a(n) = number of possibilities, allowing turning over.
Also number of 2 X 2 matrices with entries mod n, up to row and column permutation. Number of k X l matrices with entries mod n, up to row and column permutation is Z(S_k X S_l; n,n,...) where Z(S_k X S_l; x_1,x_2,...) is cycle index of Cartesian product of symmetric groups S_k and S_l of degree k and l, respectively. - Vladeta Jovovic, Nov 04 2000
Also, if a 2-set Y and a 3-set Z are disjoint subsets of an n-set X then a(n-5) is the number of 6-subsets of X intersecting both Y and Z. - Milan Janjic, Sep 08 2007
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LINKS
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FORMULA
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G.f.: (1 + 2*x + 2*x^2 + x^3)/(1 - x)^5.
a(1)=1, a(2)=7, a(3)=27, a(4)=76, a(5)=175; for n>5, a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). (End)
E.g.f.: x*(4 + 10*x + 6*x^2 + x^3)*exp(x)/4. - Ilya Gutkovskiy, Apr 16 2016
Sum_{n>=1} 1/a(n) = 2*(1 + Pi^2 - sqrt(3)*Pi*coth(sqrt(3)*Pi))/9. - Amiram Eldar, Feb 13 2023
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MAPLE
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MATHEMATICA
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Table[(n^2 (n^2+3))/4, {n, 40}] (* or *) LinearRecurrence[{5, -10, 10, -5, 1}, {1, 7, 27, 76, 175}, 40] (* Harvey P. Dale, Oct 01 2011 *)
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PROG
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(PARI) Vec((-1-2*x-2*x^2-x^3)/(x-1)^5 + O(x^50)) \\ Michel Marcus, Aug 23 2015
(PARI) a(n) = (1/4)*n^2*(n^2+3); \\ Altug Alkan, Apr 16 2016
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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Christian Meland (christian.meland(AT)pfi.no)
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EXTENSIONS
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STATUS
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approved
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