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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. 4
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 (list; graph; refs; listen; history; text; internal format)
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 A027783

Adjacent sequences:  A283113 A283114 A283115 * A283117 A283118 A283119

KEYWORD

nonn,easy

AUTHOR

Heinrich Ludwig, Mar 01 2017

STATUS

approved

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Last modified June 19 05:33 EDT 2019. Contains 324218 sequences. (Running on oeis4.)