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A289223 Number of ways to select 2 disjoint point triples from an n X n X n triangular point grid, each point triple forming an 2 X 2 X 2 triangle.

%I A289223 #11 Jul 01 2017 13:16:55
%S A289223 0,0,12,66,204,480,960,1722,2856,4464,6660,9570,13332,18096,24024,
%T A289223 31290,40080,50592,63036,77634,94620,114240,136752,162426,191544,
%U A289223 224400,261300,302562,348516,399504,455880,518010,586272,661056,742764,831810,928620,1033632,1147296
%N A289223 Number of ways to select 2 disjoint point triples from an n X n X n triangular point grid, each point triple forming an 2 X 2 X 2 triangle.
%C A289223 Rotations and reflections of a selection are regarded as different. For the number of congruence classes see A117662(n-1).
%H A289223 Heinrich Ludwig, <a href="/A289223/b289223.txt">Table of n, a(n) for n = 2..100</a>
%H A289223 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A289223 a(n) = (n^4 -4*n^3 -7*n^2 +46*n -48)/2 for n>=2.
%F A289223 From _Colin Barker_, Jun 28 2017: (Start)
%F A289223 G.f.: 6*x^4*(2 - x)*(1 + x) / (1 - x)^5.
%F A289223 a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>6.
%F A289223 (End)
%e A289223 There are 12 ways to choose two 2 X 2 X 2 triangles (xxx) from a 4 X 4 X 4 point grid, for example:
%e A289223       x           x          x
%e A289223      x x         x x        x x
%e A289223     . x x       x . .      . x .
%e A289223    . . x .     x x . .    . x x .
%e A289223 The other nine selections are reflections or rotations of these three.
%o A289223 (PARI) Vec(6*x^4*(2 - x)*(1 + x) / (1 - x)^5 + O(x^60)) \\ _Colin Barker_, Jun 28 2017
%Y A289223 Cf. A117662, A289222, A289224, A289225, A289226, A289227, A289228.
%K A289223 nonn,easy
%O A289223 2,3
%A A289223 _Heinrich Ludwig_, Jun 28 2017