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.

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

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