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

Author

Heinrich Ludwig, Mar 01 2017

Keywords

• nonn
• easy

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.

Examples

			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 . .
		

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).

Crossrefs

Cf. A193982, A283113, A283114 (3 points), A283116 (5 points).

Programs

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 *)

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

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