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.

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

Heinrich Ludwig, Table of n, a(n) for n = 5..100

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

Crossrefs

Cf. A289222, A289223, A289224, A289225, A289227, A289228, A289232.

Sequence in context: A166784 A223365 A288071 * A133712 A058834 A022046

Adjacent sequences: A289223 A289224 A289225 * A289227 A289228 A289229

Keyword

nonn,easy

Author

Heinrich Ludwig, Jul 01 2017

Status

approved