A289224 Number of 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.
0, 4, 82, 670, 3028, 9780, 25574, 57862, 117800, 221268, 390010, 652894, 1047292, 1620580, 2431758, 3553190, 5072464, 7094372, 9743010, 13163998, 17526820, 23027284, 29890102, 38371590, 48762488, 61390900, 76625354, 94877982, 116607820, 142324228, 172590430, 208027174
Offset: 3
Examples
There are four ways to choose three 2 X 2 X 2 triangles (aaa, bbb, ccc) from a 4 X 4 X 4 point grid, for example:
a a
a a a a
b c c b . c
b b c . b b c c
The other 2 possible selections are rotations of the first one.
Note: aaa, bbb, ccc 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 = 3..100
- Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).
Programs
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Maple
A289224:=n->(n^6-6*n^5-24*n^4+208*n^3-67*n^2-1684*n+2712)/6: seq(A289224(n), n=3..50); # Wesley Ivan Hurt, Jun 28 2017
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Mathematica
Table[(n^6 - 6 n^5 - 24 n^4 + 208 n^3 - 67 n^2 - 1684 n + 2712)/6, {n, 3, 34}] (* or *) LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {0, 4, 82, 670, 3028, 9780, 25574}, 32] (* or *) Drop[#, 3] &@ CoefficientList[Series[2 x^4*(2 + 27 x + 90 x^2 - 40 x^3 - 38 x^4 + 19 x^5)/(1 - x)^7, {x, 0, 34}], x] (* Michael De Vlieger, Jun 29 2017 *) -
PARI
a(n) = (n^6 - 6*n^5 - 24*n^4 + 208*n^3 - 67*n^2 - 1684*n + 2712)/6 \\ Charles R Greathouse IV, Jun 28 2017
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PARI
concat(0, Vec(2*x^4*(2 + 27*x + 90*x^2 - 40*x^3 - 38*x^4 + 19*x^5) / (1 - x)^7 + O(x^60))) \\ Colin Barker, Jun 29 2017
Formula
a(n) = (n^6 -6*n^5 -24*n^4 +208*n^3 -67*n^2 -1684*n +2712)/6.
From Colin Barker, Jun 29 2017: (Start)
G.f.: 2*x^4*(2 + 27*x + 90*x^2 - 40*x^3 - 38*x^4 + 19*x^5) / (1 - x)^7.
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n>9.
(End)
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