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

Author

Heinrich Ludwig, Jun 28 2017

Keywords

• nonn
• easy

Comments

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

Examples

			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.
		

Links

• Heinrich Ludwig, Table of n, a(n) for n = 2..100
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

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

Programs

PARI

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

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)