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 A283115 Number of nonequivalent ways (mod D_3) to place 4 points on an n X n X n triangular grid so that no two of them are on the same row, column or diagonal. 4
 0, 0, 0, 0, 0, 3, 40, 242, 1038, 3504, 9998, 25158, 57410, 121023, 239148, 447552, 799764, 1373400, 2278290, 3666036, 5742396, 8781111, 13141326, 19287246, 27811906, 39463424, 55177122, 76109826, 103681214, 139618479, 186008654, 245354424, 320640264, 415401264 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,6 COMMENTS In terms of triangular chess: Number of nonequivalent ways (mod D_3) to arrange 4 nonattacking rooks on an n X n X n board. Reflections and rotations of placements are not counted. For numbers if they are to be counted see A193982. LINKS Heinrich Ludwig, Table of n, a(n) for n = 1..1000 Index entries for linear recurrences with constant coefficients, signature (3,0,-6,0,6,8,-12,-9,13,6,-6,-13,9,12,-8,-6,0,6,0,-3,1). FORMULA a(n) = (5*n^8 - 100*n^7 + 810*n^6 - 3336*n^5 + 6940*n^4 - 5120*n^3 - 4080*n^2 + 6336*n)/11520 + IF(MOD(n, 2) = 1, 4*n^3 - 38*n^2 + 144*n - 207)/768 + IF(MOD(n, 3) = 1, n^2 - 6*n + 8)/18 + IF(MOD(n, 6) = 1, -1)/6. G.f.: x^6*(3 + 31*x + 122*x^2 + 330*x^3 + 630*x^4 + 920*x^5 + 1128*x^6 + 1224*x^7 + 1124*x^8 + 924*x^9 + 644*x^10 + 336*x^11 + 117*x^12 + 27*x^13) / ((1 - x)^9*(1 + x)^4*(1 - x + x^2)*(1 + x + x^2)^3). - Colin Barker, Mar 01 2017 EXAMPLE There are a(6) = 3 ways to place 4 points on an 6 X 6 X 6 grid, rotations and reflections ignored: . X . . X . . X . . . . . . . . . X . . X . . . X . . . . . X . . . . . X . . . . X . . . . . . X . . . . . X . . . . . X . . MATHEMATICA Table[(5 n^8 - 100 n^7 + 810 n^6 - 3336 n^5 + 6940 n^4 - 5120 n^3 - 4080 n^2 + 6336 n)/11520 + Boole[OddQ@ n] (4 n^3 - 38 n^2 + 144 n - 207)/768 + Boole[Mod[n, 3] == 1] (n^2 - 6 n + 8)/18 - Boole[Mod[n, 6] == 1]/6, {n, 34}] (* or *) Rest@ CoefficientList[Series[x^6*(3 + 31 x + 122 x^2 + 330 x^3 + 630 x^4 + 920 x^5 + 1128 x^6 + 1224 x^7 + 1124 x^8 + 924 x^9 + 644 x^10 + 336 x^11 + 117 x^12 + 27 x^13)/((1 - x)^9*(1 + x)^4*(1 - x + x^2) (1 + x + x^2)^3), {x, 0, 34}], x] (* Michael De Vlieger, Mar 01 2017 *) PROG (PARI) concat(vector(5), Vec(x^6*(3 + 31*x + 122*x^2 + 330*x^3 + 630*x^4 + 920*x^5 + 1128*x^6 + 1224*x^7 + 1124*x^8 + 924*x^9 + 644*x^10 + 336*x^11 + 117*x^12 + 27*x^13) / ((1 - x)^9*(1 + x)^4*(1 - x + x^2)*(1 + x + x^2)^3) + O(x^40))) \\ Colin Barker, Mar 01 2017 CROSSREFS Cf. A193982, A283113, A283114 (3 points), A283116 (5 points). Sequence in context: A160659 A197247 A065815 * A013257 A013250 A013255 Adjacent sequences: A283112 A283113 A283114 * A283116 A283117 A283118 KEYWORD nonn,easy AUTHOR Heinrich Ludwig, Mar 01 2017 STATUS approved

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Last modified December 6 22:16 EST 2023. Contains 367616 sequences. (Running on oeis4.)