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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.
7
0, 0, 12, 66, 204, 480, 960, 1722, 2856, 4464, 6660, 9570, 13332, 18096, 24024, 31290, 40080, 50592, 63036, 77634, 94620, 114240, 136752, 162426, 191544, 224400, 261300, 302562, 348516, 399504, 455880, 518010, 586272, 661056, 742764, 831810, 928620, 1033632, 1147296
OFFSET
2,3
COMMENTS
Rotations and reflections of a selection are regarded as different. For the number of congruence classes see A117662(n-1).
FORMULA
a(n) = (n^4 -4*n^3 -7*n^2 +46*n -48)/2 for n>=2.
From Colin Barker, Jun 28 2017: (Start)
G.f.: 6*x^4*(2 - x)*(1 + x) / (1 - x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>6.
(End)
EXAMPLE
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:
x x x
x x x x x x
. x x x . . . x .
. . x . x x . . . x x .
The other nine selections are reflections or rotations of these three.
PROG
(PARI) Vec(6*x^4*(2 - x)*(1 + x) / (1 - x)^5 + O(x^60)) \\ Colin Barker, Jun 28 2017
KEYWORD
nonn,easy
AUTHOR
Heinrich Ludwig, Jun 28 2017
STATUS
approved