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%I #9 Mar 07 2017 00:46:43
%S 0,0,0,0,0,0,1,45,533,3546,16917,64345,207327,587922,1505924,3549610,
%T 7806420,16188690,31919658,60238044,109392480,192015912,326985561,
%U 541900545,876326275,1385991432,2148140345,3268293567,4888684275,7198705228,10447710630,14960606226
%N 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.
%C 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.
%C Reflections and rotations of placements are not counted. For numbers if they are to be counted see A193983.
%H Heinrich Ludwig, <a href="/A283116/b283116.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_18">Index entries for linear recurrences with constant coefficients</a>, signature (5,-5,-14,30,6,-50,10,44,0,-44,-10,50,-6,-30,14,5,-5,1).
%F 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.
%F 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
%e There is a(7) = 1 way to place 5 points on a 7 X 7 X 7 grid, rotations and reflections ignored:
%e .
%e . .
%e . X .
%e . . . X
%e X . . . .
%e . . X . . .
%e . . . . X . .
%t 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 *)
%t 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 *)
%o (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
%Y Cf. A193983, A283113, A283114 (3 points), A283115 (4 points).
%K nonn,easy
%O 1,8
%A _Heinrich Ludwig_, Mar 01 2017
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