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A289225 Number of ways to select 4 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 A289225 #9 Jun 30 2017 21:18:31
%S A289225 0,13,859,9585,56520,231635,749223,2051819,4965960,10924065,22268395,
%T A289225 42654733,77575104,135020535,226306535,367085655,578573168,889013589,
%U A289225 1335417435,1965599305,2840550040,4037177403,5651451399,7801992035,10634139000,14324544425,19086331563
%N A289225 Number of ways to select 4 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 A289225 Rotations and reflections of a selection are regarded as different. For the number of congruence classes see A289231.
%H A289225 Heinrich Ludwig, <a href="/A289225/b289225.txt">Table of n, a(n) for n = 4..100</a>
%H A289225 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (9,-36,84,-126,126,-84,36,-9,1).
%F A289225 a(n) = (n^8 -8*n^7 -50*n^6 +556*n^5 +231*n^4 -12388*n^3 +17914*n^2 +86648*n -198528)/24.
%F A289225 From _Colin Barker_, Jun 30 2017: (Start)
%F A289225 G.f.: x^5*(13 + 742*x + 2322*x^2 + 87*x^3 - 2503*x^4 + 684*x^5 + 560*x^6 - 225*x^7) / (1 - x)^9.
%F A289225 a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9) for n>12.
%F A289225 (End)
%e A289225 There are thirteen ways to choose four 2 X 2 X 2 triangles (aaa, ..., ddd) from a 5 X 5 X 5 point grid, for example:
%e A289225       a           a           a           .
%e A289225      a a         a a         a a         a a
%e A289225     b c c       . d .       . . .       . a .
%e A289225    b b c d     b d d c     b c c d     b c c d
%e A289225   . . . d d   b b . c c   b b c d d   b b c d d
%e A289225 The other nine possible selections are rotations and reflections of these.
%e A289225 Note: aaa, ..., ddd are not distinguishable, they are denoted differently for a better perception of the 2 X 2 X 2 triangles only.
%p A289225 A289225:=n->(n^8 -8*n^7 -50*n^6 +556*n^5 +231*n^4 -12388*n^3 +17914*n^2 +86648*n -198528)/24: seq(A289225(n), n=4..50); # _Wesley Ivan Hurt_, Jun 29 2017
%o A289225 (PARI) concat(0, Vec(x^5*(13 + 742*x + 2322*x^2 + 87*x^3 - 2503*x^4 + 684*x^5 + 560*x^6 - 225*x^7) / (1 - x)^9 + O(x^30))) \\ _Colin Barker_, Jun 30 2017
%Y A289225 Cf. A289222, A289223, A289224, A289226, A289227, A289228, A289231.
%K A289225 nonn,easy
%O A289225 4,2
%A A289225 _Heinrich Ludwig_, Jun 29 2017