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A289226
Number of 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.
8
0, 420, 15108, 190371, 1336320, 6528948, 24951780, 79851975, 223419840, 562591836, 1301255556, 2806131075, 5705746752, 11034449244, 20436317412, 36447218199, 62877079680, 105318792564, 171815016708, 273719593923, 426796282752, 652604165220, 980226360036, 1448406641607
OFFSET
5,2
COMMENTS
Rotations and reflections of a selection are regarded as different. For the number of congruence classes see A289232.
LINKS
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).
FORMULA
a(n) = (n^10 -10*n^9 -85*n^8 +1160*n^7 +1345*n^6 -49162*n^5 +62145*n^4 +892140*n^3 -2198566*n^2 -5725008*n +18190440)/120.
G.f.: 3*x^6*(140 + 3496*x + 15761*x^2 + 1293*x^3 - 18129*x^4 + 3779*x^5 + 6103*x^6 - 1637*x^7 - 1139*x^8 + 413*x^9) / (1 - x)^11. - Colin Barker, Jul 01 2017
EXAMPLE
There are 420 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.
PROG
(PARI) concat(0, Vec(3*x^6*(140 + 3496*x + 15761*x^2 + 1293*x^3 - 18129*x^4 + 3779*x^5 + 6103*x^6 - 1637*x^7 - 1139*x^8 + 413*x^9) / (1 - x)^11 + O(x^40))) \\ Colin Barker, Jul 01 2017
KEYWORD
nonn,easy
AUTHOR
Heinrich Ludwig, Jul 01 2017
STATUS
approved