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A289232 Number of nonequivalent ways to select 5 disjoint point triples from an n X n X n triangular point grid, each point triple forming a 2 X 2 X 2 triangle.

%I A289232 #8 Jul 01 2017 13:15:49
%S A289232 0,77,2569,31951,223346,1089665,4161705,13314461,37246668,93781829,
%T A289232 216901737,467727523,951014654,1839155785,3406165049,6074688977,
%U A289232 10479716856,17553399741,28636182537,45620375447,71133273514,108768061009,163371926729,241402171109,351362501892
%N A289232 Number of nonequivalent ways to select 5 disjoint point triples from an n X n X n triangular point grid, each point triple forming a 2 X 2 X 2 triangle.
%C A289232 Rotations and reflections of a selection are not counted. If they are to be counted see A289226.
%H A289232 Heinrich Ludwig, <a href="/A289232/b289232.txt">Table of n, a(n) for n = 5..100</a>
%H A289232 <a href="/index/Rec#order_14">Index entries for linear recurrences with constant coefficients</a>, signature (8,-25,32,11,-88,99,0,-99,88,-11,-32,25,-8,1).
%F A289232 a(n) = (n^10 -10*n^9 -85*n^8 +1160*n^7 +1345*n^6 -49084*n^5 +61035*n^4 +897210*n^3 -2205196*n^2 -5725656*n +18174960)/720 + IF(MOD(n, 2) = 1, -2*n^2 +13*n -11)/4.
%F A289232 G.f.: x^6*(77 + 1953*x + 13324*x^2 + 29499*x^3 + 18617*x^4 - 15880*x^5 - 17638*x^6 + 4876*x^7 + 8057*x^8 - 881*x^9 - 1966*x^10 + 81*x^11 + 201*x^12) / ((1 - x)^11*(1 + x)^3). - _Colin Barker_, Jul 01 2017
%e A289232 There are 77 nonequivalent ways to choose five 2 X 2 X 2 triangles (aaa, ..., eee) from a 6 X 6 X 6 point grid, for example:
%e A289232         .               a
%e A289232        . .             a a
%e A289232       . . .           . d .
%e A289232      a a b b         b d d c
%e A289232     c a d b e       b b e c c
%e A289232    c c d d e e     . . e e . .
%e A289232 Note: aaa, ..., eee are not distinguishable, they are denoted differently for a better perception of the 2 X 2 X 2 triangles only.
%o A289232 (PARI) concat(0, Vec(x^6*(77 + 1953*x + 13324*x^2 + 29499*x^3 + 18617*x^4 - 15880*x^5 - 17638*x^6 + 4876*x^7 + 8057*x^8 - 881*x^9 - 1966*x^10 + 81*x^11 + 201*x^12) / ((1 - x)^11*(1 + x)^3) + O(x^40))) \\ _Colin Barker_, Jul 01 2017
%Y A289232 Cf. A289229, A117662, A289230, A289231, A289226.
%K A289232 nonn,easy
%O A289232 5,2
%A A289232 _Heinrich Ludwig_, Jul 01 2017