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A240440
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Number of ways to place 3 points on a triangular grid of side n so that they are not vertices of an equilateral triangle of any orientation.
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6
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0, 0, 15, 105, 420, 1260, 3150, 6930, 13860, 25740, 45045, 75075, 120120, 185640, 278460, 406980, 581400, 813960, 1119195, 1514205, 2018940, 2656500, 3453450, 4440150, 5651100, 7125300, 8906625, 11044215, 13592880, 16613520, 20173560, 24347400, 29216880
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OFFSET
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1,3
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COMMENTS
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a(n) = A001498(n,3), the fourth column of coefficients of Bessel polynomials. - Ran Pan, Dec 03 2015
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LINKS
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S. Butler, P. Karasik, A note on nested sums, J. Int. Seq. 13 (2010), 10.4.4, p=5 in the last equation on page 3.
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FORMULA
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a(n) = (n+3)*(n+2)*(n+1)*n*(n-1)*(n-2)/48.
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>7. - Wesley Ivan Hurt, Dec 03 2015
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MAPLE
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MATHEMATICA
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Table[(n+3)(n+2)(n+1)n(n-1)(n-2)/48, {n, 50}] (* Wesley Ivan Hurt, Apr 08 2014 *)
CoefficientList[Series[15 x^2/(1 - x)^7, {x, 0, 40}], x] (* Vincenzo Librandi, Apr 19 2014 *)
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PROG
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(PARI) Vec(15*x^3/(1-x)^7 + O(x^100)) \\ Colin Barker, Apr 18 2014
(Magma) [(n+3)*(n+2)*(n+1)*n*(n-1)*(n-2)/48 : n in [1..50]]; // Wesley Ivan Hurt, Dec 03 2015
(PARI) vector(100, n, (n^2-1)*(n^2-4)*(n+3)*n/48) \\ Derek Orr, Dec 24 2015
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CROSSREFS
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If one of the initial zeros is omitted, this is a row of the array in A129533.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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