login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

A289230
Number of nonequivalent ways to select 3 disjoint point triples from an n X n X n triangular point grid, each point triple forming a 2 X 2 X 2 triangle.
5
0, 2, 19, 127, 536, 1688, 4357, 9789, 19844, 37172, 65397, 109335, 175214, 270934, 406329, 593463, 846934, 1184212, 1625979, 2196509, 2924050, 3841240, 4985531, 6399647, 8132044, 10237410, 12777167, 15820007, 19442436, 23729352, 28774625, 34681717, 41564304, 49546932
OFFSET
3,2
COMMENTS
Rotations and reflections of a selection are not counted. If they are to be counted see A289224.
LINKS
FORMULA
a(n) = (n^6 -6*n^5 -24*n^4 +220*n^3 -153*n^2 -1488*n +2592)/36 + IF(MOD(n, 2) = 1, -1)/2 + IF(MOD(n, 3) = 1, -2)/9.
G.f.: x^4*(2 + 9*x + 50*x^2 + 60*x^3 + 37*x^4 - 21*x^5 - 20*x^6 - 4*x^7 + 7*x^8) / ((1 - x)^7*(1 + x)*(1 + x + x^2)). - Colin Barker, Jun 30 2017
EXAMPLE
There are two nonequivalent ways to choose three 2 X 2 X 2 triangles (aaa, bbb, ccc) from a 4 X 4 X 4 point grid:
a a
a a a a
b c c b . c
b b c . b b c c
Note: aaa, bbb, ccc 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^4*(2 + 9*x + 50*x^2 + 60*x^3 + 37*x^4 - 21*x^5 - 20*x^6 - 4*x^7 + 7*x^8) / ((1 - x)^7*(1 + x)*(1 + x + x^2)) + O(x^40))) \\ Colin Barker, Jun 30 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Heinrich Ludwig, Jun 30 2017
STATUS
approved