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A289227 Number of ways to select 6 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 A289227 #7 Jul 01 2017 08:26:15
%S A289227 0,76,10956,371016,4988324,39302784,218633416,952344088,3460482612,
%T A289227 10932805668,30901640212,79762409256,190898410020,428596770008,
%U A289227 910935932112,1846146025240,3588666200596,6723331905852,12188915557404,21455723224456,36776237135268,61533021405936
%N A289227 Number of ways to select 6 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 A289227 Rotations and reflections of a selection are regarded as different.
%H A289227 Heinrich Ludwig, <a href="/A289227/b289227.txt">Table of n, a(n) for n = 5..100</a>
%H A289227 <a href="/index/Rec#order_13">Index entries for linear recurrences with constant coefficients</a>, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).
%F A289227 a(n) = (n^12 -12*n^11 -129*n^10 +2090*n^9 +3985*n^8 -142832*n^7 +152809*n^6 +4752598*n^5 -12392266*n^4 -76011076*n^3 +274393360*n^2 +455879232*n -2015187840)/720 for n>=6.
%F A289227 G.f.: 4*x^6*(19 + 2492*x + 58629*x^2 + 249487*x^3 + 78686*x^4 - 397088*x^5 + 93163*x^6 + 160960*x^7 - 77014*x^8 - 10728*x^9 + 4312*x^10 + 5013*x^11 - 1611*x^12) / (1 - x)^13. - _Colin Barker_, Jul 01 2017
%e A289227 There are 76 ways to choose six 2 X 2 X 2 triangles (aaa, ..., fff) from a 6 X 6 X 6 point grid, for example:
%e A289227         a               a
%e A289227        a a             a a
%e A289227       . . .           b . c
%e A289227      b b c c         b b c c
%e A289227     d b e c f       d . e . f
%e A289227    d d e e f f     d d e e f f
%e A289227 Note: aaa, ..., fff are not distinguishable, they are denoted differently for a better perception of the 2 X 2 X 2 triangles only.
%o A289227 (PARI) concat(0, Vec(4*x^6*(19 + 2492*x + 58629*x^2 + 249487*x^3 + 78686*x^4 - 397088*x^5 + 93163*x^6 + 160960*x^7 - 77014*x^8 - 10728*x^9 + 4312*x^10 + 5013*x^11 - 1611*x^12) / (1 - x)^13 + O(x^40))) \\ _Colin Barker_, Jul 01 2017
%Y A289227 Cf. A289222, A289223, A289224, A289225, A289226, A289228.
%K A289227 nonn,easy
%O A289227 5,2
%A A289227 _Heinrich Ludwig_, Jul 01 2017