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A283114 Number of nonequivalent ways (mod D_3) to place 3 points on an n X n X n triangular grid so that no two of them are on the same row, column or diagonal.

%I A283114 #22 May 07 2022 10:50:00
%S A283114 0,0,0,1,5,23,82,230,560,1208,2392,4405,7673,12733,20320,31326,46914,
%T A283114 68460,97698,136635,187737,253813,338240,444818,578038,742898,945224,
%U A283114 1191443,1488955,1845865,2271410,2775640,3369910,4066506,4879200,5822823,6913887,8170095
%N A283114 Number of nonequivalent ways (mod D_3) to place 3 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 A283114 In terms of triangular chess: Number of nonequivalent ways (mod D_3) to arrange 3 nonattacking rooks on an n X n X n board.
%C A283114 Reflections and rotations of placements are not counted. For numbers if they are to be counted see A193981.
%H A283114 Heinrich Ludwig, <a href="/A283114/b283114.txt">Table of n, a(n) for n = 1..1000</a>
%H A283114 <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (3,0,-7,3,6,0,-6,-3,7,0,-3,1).
%F A283114 a(n) = (n^6 - 9*n^5 + 27*n^4 - 20*n^3 - 24*n^2 + 24*n)/288 + IF(MOD(n, 2) = 1, n^2 - 3*n - 5)/32 + IF(MOD(n, 3) = 1, 2)/9.
%F A283114 G.f.: x^4*(1 + 2*x + 8*x^2 + 20*x^3 + 16*x^4 + 10*x^5 + 3*x^6) / ((1 - x)^7*(1 + x)^3*(1 + x + x^2)). - _Colin Barker_, Mar 01 2017
%F A283114 a(n) = ( 2*n^6 - 18*n^5 + 54*n^4 - 40*n^3 - 39*n^2 + 21*n - 45 - 9*(n^2 - 3*n - 5)*(-1)^n + 128*((n mod 3) mod 2) )/576. - _Bruno Berselli_, Mar 01 2017
%e A283114 There is a(4) = 1 way to place 3 points on a 4 X 4 X 4 grid, rotations and reflections ignored:
%e A283114      .
%e A283114     X .
%e A283114    . . X
%e A283114   . X . .
%t A283114 Table[(n^6 - 9 n^5 + 27 n^4 - 20 n^3 - 24 n^2 + 24 n)/288 + Boole[OddQ@ n] (n^2 - 3 n - 5)/32 + Boole[Mod[n, 3] == 1] 2/9, {n, 38}] (* or *)
%t A283114 Rest@ CoefficientList[Series[x^4*(1 + 2 x + 8 x^2 + 20 x^3 + 16 x^4 + 10 x^5 + 3 x^6)/((1 - x)^7*(1 + x)^3*(1 + x + x^2)), {x, 0, 38}], x] (* _Michael De Vlieger_, Mar 01 2017 *)
%t A283114 LinearRecurrence[{3,0,-7,3,6,0,-6,-3,7,0,-3,1},{0,0,0,1,5,23,82,230,560,1208,2392,4405},40] (* _Harvey P. Dale_, May 07 2022 *)
%o A283114 (PARI) concat(vector(3), Vec(x^4*(1 + 2*x + 8*x^2 + 20*x^3 + 16*x^4 + 10*x^5 + 3*x^6) / ((1 - x)^7*(1 + x)^3*(1 + x + x^2)) + O(x^30))) \\ _Colin Barker_, Mar 01 2017
%Y A283114 Cf. A193981, A283113, A283115 (4 points), A283116 (5 points).
%K A283114 nonn,easy
%O A283114 1,5
%A A283114 _Heinrich Ludwig_, Mar 01 2017