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 A240440 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. 6
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS a(n) = 15 * A000579(n+3). a(n) = A001498(n,3), the fourth column of coefficients of Bessel polynomials. - Ran Pan, Dec 03 2015 LINKS Vincenzo Librandi, Table of n, a(n) for n = 1..1000 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. Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1). FORMULA a(n) = (n+3)*(n+2)*(n+1)*n*(n-1)*(n-2)/48. G.f.: 15*x^3 / (1-x)^7. - Colin Barker, Apr 18 2014 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 MAPLE A240440:=n->(n+3)*(n+2)*(n+1)*n*(n-1)*(n-2)/48; seq(A240440(n), n=1..50); # Wesley Ivan Hurt, Apr 08 2014 MATHEMATICA 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 *) PROG (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 CROSSREFS Cf. A000579, A001498, A240441, A240442, A240439. If one of the initial zeros is omitted, this is a row of the array in A129533. Sequence in context: A174385 A185129 A090454 * A256287 A047640 A296912 Adjacent sequences: A240437 A240438 A240439 * A240441 A240442 A240443 KEYWORD nonn,easy AUTHOR Heinrich Ludwig, Apr 08 2014 STATUS approved

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Last modified June 2 22:56 EDT 2023. Contains 363102 sequences. (Running on oeis4.)