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A289232
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Number of nonequivalent ways to select 5 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, 77, 2569, 31951, 223346, 1089665, 4161705, 13314461, 37246668, 93781829, 216901737, 467727523, 951014654, 1839155785, 3406165049, 6074688977, 10479716856, 17553399741, 28636182537, 45620375447, 71133273514, 108768061009, 163371926729, 241402171109, 351362501892
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OFFSET
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5,2
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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 A289226.
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (8,-25,32,11,-88,99,0,-99,88,-11,-32,25,-8,1).
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FORMULA
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a(n) = (n^10 -10*n^9 -85*n^8 +1160*n^7 +1345*n^6 -49084*n^5 +61035*n^4 +897210*n^3 -2205196*n^2 -5725656*n +18174960)/720 + IF(MOD(n, 2) = 1, -2*n^2 +13*n -11)/4.
G.f.: x^6*(77 + 1953*x + 13324*x^2 + 29499*x^3 + 18617*x^4 - 15880*x^5 - 17638*x^6 + 4876*x^7 + 8057*x^8 - 881*x^9 - 1966*x^10 + 81*x^11 + 201*x^12) / ((1 - x)^11*(1 + x)^3). - Colin Barker, Jul 01 2017
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EXAMPLE
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There are 77 nonequivalent ways to choose five 2 X 2 X 2 triangles (aaa, ..., eee) from a 6 X 6 X 6 point grid, for example:
. a
. . a a
. . . . d .
a a b b b d d c
c a d b e b b e c c
c c d d e e . . e e . .
Note: aaa, ..., eee 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^6*(77 + 1953*x + 13324*x^2 + 29499*x^3 + 18617*x^4 - 15880*x^5 - 17638*x^6 + 4876*x^7 + 8057*x^8 - 881*x^9 - 1966*x^10 + 81*x^11 + 201*x^12) / ((1 - x)^11*(1 + x)^3) + O(x^40))) \\ Colin Barker, Jul 01 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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