A289227 Number of ways to select 6 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, 76, 10956, 371016, 4988324, 39302784, 218633416, 952344088, 3460482612, 10932805668, 30901640212, 79762409256, 190898410020, 428596770008, 910935932112, 1846146025240, 3588666200596, 6723331905852, 12188915557404, 21455723224456, 36776237135268, 61533021405936

Offset: 5

Author

Heinrich Ludwig, Jul 01 2017

Keywords

• nonn
• easy

Comments

Rotations and reflections of a selection are regarded as different.

Examples

			There are 76 ways to choose six 2 X 2 X 2 triangles (aaa, ..., fff) from a 6 X 6 X 6 point grid, for example:
        a               a
       a a             a a
      . . .           b . c
     b b c c         b b c c
    d b e c f       d . e . f
   d d e e f f     d d e e f f
Note: aaa, ..., fff 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 = 5..100
• Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Crossrefs

Cf. A289222, A289223, A289224, A289225, A289226, A289228.

Programs

PARI

concat(0, Vec(4*x^6*(19 + 2492*x + 58629*x^2 + 249487*x^3 + 78686*x^4 - 397088*x^5 + 93163*x^6 + 160960*x^7 - 77014*x^8 - 10728*x^9 + 4312*x^10 + 5013*x^11 - 1611*x^12) / (1 - x)^13 + O(x^40))) \\ Colin Barker, Jul 01 2017

Formula

a(n) = (n^12 -12*n^11 -129*n^10 +2090*n^9 +3985*n^8 -142832*n^7 +152809*n^6 +4752598*n^5 -12392266*n^4 -76011076*n^3 [phone]*n^2 [phone]*n -2015187840)/720 for n>=6.

G.f.: 4*x^6*(19 + 2492*x + 58629*x^2 + 249487*x^3 + 78686*x^4 - 397088*x^5 + 93163*x^6 + 160960*x^7 - 77014*x^8 - 10728*x^9 + 4312*x^10 + 5013*x^11 - 1611*x^12) / (1 - x)^13. - Colin Barker, Jul 01 2017