A289231 - OEIS

%I #12 Jul 01 2017 13:17:20

%S 0,4,159,1644,9548,38872,125367,342831,829052,1822785,3714519,7113539,

%T 12935256,22511616,37728563,61194888,96446684,148191316,222597315,

%U 327633979,473466444,672912717,941968139,1300402591,1772439504,2387521212,3181168199,4195941108,5482512012

%N Number of nonequivalent ways to select 4 disjoint point triples from an n X n X n triangular point grid, each point triple forming a 2 X 2 X 2 triangle.

%C Rotations and reflections of a selection are not counted. If they are to be counted see A289225.

%H Heinrich Ludwig, <a href="/A289231/b289231.txt">Table of n, a(n) for n = 4..100</a>

%H <a href="/index/Rec#order_17">Index entries for linear recurrences with constant coefficients</a>, signature (4,-4,-1,-2,8,5,-14,-1,1,14,-5,-8,2,1,4,-4,1).

%F a(n) = (n^8 -8*n^7 -50*n^6 +556*n^5 +261*n^4 -12724*n^3 +19088*n^2 +86016*n -201024)/144 + IF(MOD(n, 2) = 1, -2*n +5)/4 + IF(MOD(n, 3) = 1, -n^2 +2*n +12)/9.

%F G.f.: x^5*(4 + 143*x + 1024*x^2 + 3612*x^3 + 7423*x^4 + 10001*x^5 + 8395*x^6 + 3273*x^7 - 1362*x^8 - 2393*x^9 - 878*x^10 + 488*x^11 + 539*x^12 + 101*x^13 - 89*x^14 - 41*x^15) / ((1 - x)^9*(1 + x)^2*(1 + x + x^2)^3). - _Colin Barker_, Jun 30 2017

%e There are four nonequivalent ways to choose four 2 X 2 X 2 triangles (aaa, ..., ddd) from a 5 X 5 X 5 point grid:

%e       a           a           a           .

%e      a a         a a         a a         a a

%e     b c c       . d .       . . .       . a .

%e    b b c d     b d d c     b c c d     b c c d

%e   . . . d d   b b . c c   b b c d d   b b c d d

%e Note: aaa, ..., ddd are not distinguishable, they are denoted differently for a better perception of the 2 X 2 X 2 triangles only.

%o (PARI) concat(0, Vec(x^5*(4 + 143*x + 1024*x^2 + 3612*x^3 + 7423*x^4 + 10001*x^5 + 8395*x^6 + 3273*x^7 - 1362*x^8 - 2393*x^9 - 878*x^10 + 488*x^11 + 539*x^12 + 101*x^13 - 89*x^14 - 41*x^15) / ((1 - x)^9*(1 + x)^2*(1 + x + x^2)^3) + O(x^40))) \\ _Colin Barker_, Jun 30 2017

%Y Cf. A289229, A289225, A117662, A289230, A289232.

%K nonn,easy

%O 4,2

%A _Heinrich Ludwig_, Jun 30 2017