A289225 Number of ways to select 4 disjoint point triples from an n X n X n triangular point grid, each point triple forming a 2 X 2 X 2 triangle.
0, 13, 859, 9585, 56520, 231635, 749223, 2051819, 4965960, 10924065, 22268395, 42654733, 77575104, 135020535, 226306535, 367085655, 578573168, 889013589, 1335417435, 1965599305, 2840550040, 4037177403, 5651451399, 7801992035, 10634139000, 14324544425, 19086331563
Offset: 4
Examples
There are thirteen ways to choose four 2 X 2 X 2 triangles (aaa, ..., ddd) from a 5 X 5 X 5 point grid, for example:
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
The other nine possible selections are rotations and reflections of these.
Note: aaa, ..., ddd are not distinguishable, they are denoted differently for a better perception of the 2 X 2 X 2 triangles only.
Links
- Heinrich Ludwig, Table of n, a(n) for n = 4..100
- Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).
Programs
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Maple
A289225:=n->(n^8 -8*n^7 -50*n^6 +556*n^5 +231*n^4 -12388*n^3 +17914*n^2 +86648*n -198528)/24: seq(A289225(n), n=4..50); # Wesley Ivan Hurt, Jun 29 2017
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PARI
concat(0, Vec(x^5*(13 + 742*x + 2322*x^2 + 87*x^3 - 2503*x^4 + 684*x^5 + 560*x^6 - 225*x^7) / (1 - x)^9 + O(x^30))) \\ Colin Barker, Jun 30 2017
Formula
a(n) = (n^8 -8*n^7 -50*n^6 +556*n^5 +231*n^4 -12388*n^3 +17914*n^2 +86648*n -198528)/24.
From Colin Barker, Jun 30 2017: (Start)
G.f.: x^5*(13 + 742*x + 2322*x^2 + 87*x^3 - 2503*x^4 + 684*x^5 + 560*x^6 - 225*x^7) / (1 - x)^9.
a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9) for n>12.
(End)
Comments