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A289231
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Number of nonequivalent 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.
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5
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0, 4, 159, 1644, 9548, 38872, 125367, 342831, 829052, 1822785, 3714519, 7113539, 12935256, 22511616, 37728563, 61194888, 96446684, 148191316, 222597315, 327633979, 473466444, 672912717, 941968139, 1300402591, 1772439504, 2387521212, 3181168199, 4195941108, 5482512012
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OFFSET
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4,2
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COMMENTS
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Rotations and reflections of a selection are not counted. If they are to be counted see A289225.
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (4,-4,-1,-2,8,5,-14,-1,1,14,-5,-8,2,1,4,-4,1).
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FORMULA
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a(n) = (n^8 -8*n^7 -50*n^6 +556*n^5 +261*n^4 -12724*n^3 +19088*n^2 +86016*n -201024)/144 + IF(MOD(n, 2) = 1, -2*n +5)/4 + IF(MOD(n, 3) = 1, -n^2 +2*n +12)/9.
G.f.: x^5*(4 + 143*x + 1024*x^2 + 3612*x^3 + 7423*x^4 + 10001*x^5 + 8395*x^6 + 3273*x^7 - 1362*x^8 - 2393*x^9 - 878*x^10 + 488*x^11 + 539*x^12 + 101*x^13 - 89*x^14 - 41*x^15) / ((1 - x)^9*(1 + x)^2*(1 + x + x^2)^3). - Colin Barker, Jun 30 2017
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EXAMPLE
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There are four nonequivalent ways to choose four 2 X 2 X 2 triangles (aaa, ..., ddd) from a 5 X 5 X 5 point grid:
a a a .
a a a a a a a a
b c c . d . . . . . a .
b b c d b d d c b c c d b c c d
. . . d d b b . c c b b c d d b b c d d
Note: aaa, ..., ddd are not distinguishable, they are denoted differently for a better perception of the 2 X 2 X 2 triangles only.
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PROG
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(PARI) concat(0, Vec(x^5*(4 + 143*x + 1024*x^2 + 3612*x^3 + 7423*x^4 + 10001*x^5 + 8395*x^6 + 3273*x^7 - 1362*x^8 - 2393*x^9 - 878*x^10 + 488*x^11 + 539*x^12 + 101*x^13 - 89*x^14 - 41*x^15) / ((1 - x)^9*(1 + x)^2*(1 + x + x^2)^3) + O(x^40))) \\ Colin Barker, Jun 30 2017
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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