A289223 Number of ways to select 2 disjoint point triples from an n X n X n triangular point grid, each point triple forming an 2 X 2 X 2 triangle.

0, 0, 12, 66, 204, 480, 960, 1722, 2856, 4464, 6660, 9570, 13332, 18096, 24024, 31290, 40080, 50592, 63036, 77634, 94620, 114240, 136752, 162426, 191544, 224400, 261300, 302562, 348516, 399504, 455880, 518010, 586272, 661056, 742764, 831810, 928620, 1033632, 1147296

Offset

2,3

Comments

Rotations and reflections of a selection are regarded as different. For the number of congruence classes see A117662(n-1).

Links

Paolo Xausa, Table of n, a(n) for n = 2..10000 (terms 2..100 from Heinrich Ludwig).

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Formula

a(n) = (n^4 -4*n^3 -7*n^2 +46*n -48)/2 for n>=2.

From Colin Barker, Jun 28 2017: (Start)

G.f.: 6*x^4*(2 - x)*(1 + x) / (1 - x)^5.

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>6. (End)

a(n) = 6 * A328152(n-1). - Alois P. Heinz, Feb 06 2026

Example

There are 12 ways to choose two 2 X 2 X 2 triangles (xxx) from a 4 X 4 X 4 point grid, for example:
      x           x          x
     x x         x x        x x
    . x x       x . .      . x .
   . . x .     x x . .    . x x .
The other nine selections are reflections or rotations of these three.

Mathematica

LinearRecurrence[{5, -10, 10, -5, 1}, {0, 0, 12, 66, 204}, 50] (* Paolo Xausa, Feb 06 2026 *)

Prog

(PARI) Vec(6*x^4*(2 - x)*(1 + x) / (1 - x)^5 + O(x^60)) \\ Colin Barker, Jun 28 2017

Crossrefs

Cf. A117662, A289222, A289224, A289225, A289226, A289227, A289228, A328152.

Sequence in context: A363591 A223234 A378742 * A296914 A285580 A001490

Adjacent sequences: A289220 A289221 A289222 * A289224 A289225 A289226

Keyword

nonn,easy

Author

Heinrich Ludwig, Jun 28 2017

Status

approved