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A289224 Number of ways to select 3 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 A289224 #18 Jul 01 2017 13:15:16
%S A289224 0,4,82,670,3028,9780,25574,57862,117800,221268,390010,652894,1047292,
%T A289224 1620580,2431758,3553190,5072464,7094372,9743010,13163998,17526820,
%U A289224 23027284,29890102,38371590,48762488,61390900,76625354,94877982,116607820,142324228,172590430,208027174
%N A289224 Number of ways to select 3 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 A289224 Rotations and reflections of a selection are regarded as different. For the number of congruence classes see A289230.
%H A289224 Heinrich Ludwig, <a href="/A289224/b289224.txt">Table of n, a(n) for n = 3..100</a>
%H A289224 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F A289224 a(n) = (n^6 -6*n^5 -24*n^4 +208*n^3 -67*n^2 -1684*n +2712)/6.
%F A289224 From _Colin Barker_, Jun 29 2017: (Start)
%F A289224 G.f.: 2*x^4*(2 + 27*x + 90*x^2 - 40*x^3 - 38*x^4 + 19*x^5) / (1 - x)^7.
%F A289224 a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n>9.
%F A289224 (End)
%e A289224 There are four ways to choose three 2 X 2 X 2 triangles (aaa, bbb, ccc) from a 4 X 4 X 4 point grid, for example:
%e A289224       a           a
%e A289224      a a         a a
%e A289224     b c c       b . c
%e A289224    b b c .     b b c c
%e A289224 The other 2 possible selections are rotations of the first one.
%e A289224 Note: aaa, bbb, ccc are not distinguishable, they are denoted differently for a better perception of the 2 X 2 X 2 triangles only.
%p A289224 A289224:=n->(n^6-6*n^5-24*n^4+208*n^3-67*n^2-1684*n+2712)/6: seq(A289224(n), n=3..50); # _Wesley Ivan Hurt_, Jun 28 2017
%t A289224 Table[(n^6 - 6 n^5 - 24 n^4 + 208 n^3 - 67 n^2 - 1684 n + 2712)/6, {n, 3, 34}] (* or *)
%t A289224 LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {0, 4, 82, 670, 3028, 9780, 25574}, 32] (* or *)
%t A289224 Drop[#, 3] &@ CoefficientList[Series[2 x^4*(2 + 27 x + 90 x^2 - 40 x^3 - 38 x^4 + 19 x^5)/(1 - x)^7, {x, 0, 34}], x] (* _Michael De Vlieger_, Jun 29 2017 *)
%o A289224 (PARI) a(n) = (n^6 - 6*n^5 - 24*n^4 + 208*n^3 - 67*n^2 - 1684*n + 2712)/6 \\ _Charles R Greathouse IV_, Jun 28 2017
%o A289224 (PARI) concat(0, Vec(2*x^4*(2 + 27*x + 90*x^2 - 40*x^3 - 38*x^4 + 19*x^5) / (1 - x)^7 + O(x^60))) \\ _Colin Barker_, Jun 29 2017
%Y A289224 Cf. A289222, A289223, A289225, A289226, A289227, A289228, A289230.
%K A289224 nonn,easy
%O A289224 3,2
%A A289224 _Heinrich Ludwig_, Jun 28 2017