A283116 Number of nonequivalent ways (mod D_3) to place 5 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, 0, 1, 45, 533, 3546, 16917, 64345, 207327, 587922, 1505924, 3549610, 7806420, 16188690, 31919658, 60238044, 109392480, 192015912, 326985561, 541900545, 876326275, 1385991432, 2148140345, 3268293567, 4888684275, 7198705228, 10447710630, 14960606226

Offset

1,8

Comments

In terms of triangular chess: Number of nonequivalent ways (mod D_3) to arrange 5 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 A193983.

Links

Heinrich Ludwig, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (5,-5,-14,30,6,-50,10,44,0,-44,-10,50,-6,-30,14,5,-5,1).

Formula

a(n) = (n^10 - 35*n^9 + 530*n^8 - 4526*n^7 + 23693*n^6 - 76544*n^5 + 141360*n^4 - 104944*n^3 - 67984*n^2 + 124224*n)/23040 + IF(MOD(n, 2) = 1, - 45*n^4 + 1350*n^3 - 13770*n^2 + 57915*n - 81225)/23040 + IF(MOD(n, 3) = 2, -2)/9.

G.f.: x^7*(1 + 40*x + 313*x^2 + 1120*x^3 + 2452*x^4 + 3596*x^5 + 3621*x^6 + 2512*x^7 + 1149*x^8 + 316*x^9) / ((1 - x)^11*(1 + x)^5*(1 + x + x^2)). - Colin Barker, Mar 01 2017

Example

There is a(7) = 1 way to place 5 points on a 7 X 7 X 7 grid, rotations and reflections ignored:
         .
        . .
       . X .
      . . . X
     X . . . .
    . . X . . .
   . . . . X . .

Mathematica

Table[(n^10 - 35 n^9 + 530 n^8 - 4526 n^7 + 23693 n^6 - 76544 n^5 + 141360 n^4 - 104944 n^3 - 67984 n^2 + 124224 n)/23040 + Boole[OddQ@ n] (-45 n^4 + 1350 n^3 - 13770 n^2 + 57915 n - 81225)/23040 - 2 Boole[Mod[n, 3] == 2]/9, {n, 32}] (* or *)
Rest@ CoefficientList[Series[x^7*(1 + 40 x + 313 x^2 + 1120 x^3 + 2452 x^4 + 3596 x^5 + 3621 x^6 + 2512 x^7 + 1149 x^8 + 316 x^9)/((1 - x)^11*(1 + x)^5*(1 + x + x^2)), {x, 0, 32}], x] (* Michael De Vlieger, Mar 01 2017 *)

Prog

(PARI) concat(vector(6), Vec(x^7*(1 + 40*x + 313*x^2 + 1120*x^3 + 2452*x^4 + 3596*x^5 + 3621*x^6 + 2512*x^7 + 1149*x^8 + 316*x^9) / ((1 - x)^11*(1 + x)^5*(1 + x + x^2)) + O(x^40))) \\ Colin Barker, Mar 01 2017

Crossrefs

Cf. A193983, A283113, A283114 (3 points), A283115 (4 points).

Sequence in context: A190417 A093529 A197501 * A223047 A147842 A349849

Adjacent sequences: A283113 A283114 A283115 * A283117 A283118 A283119

Keyword

nonn,easy

Author

Heinrich Ludwig, Mar 01 2017

Status

approved