OFFSET
4,2
COMMENTS
Rotations and reflections of a selection are not counted. If they are to be counted see A289225.
LINKS
Heinrich Ludwig, Table of n, a(n) for n = 4..100
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).
FORMULA
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
EXAMPLE
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.
PROG
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Heinrich Ludwig, Jun 30 2017
STATUS
approved