cp's OEIS Frontend

A194478 Number of ways to arrange 6 indistinguishable points on an n X n X n triangular grid so that no three points are in the same row or diagonal.

%I A194478 #16 Mar 07 2023 07:36:08
%S A194478 0,0,0,1,337,8733,96478,668028,3413828,14054915,49171641,151422970,
%T A194478 420674150,1073422309,2550004472,5699074284,12082541388,24462528078,
%U A194478 47555986746,89173692795,161899772067,285517344145,490447009030
%N A194478 Number of ways to arrange 6 indistinguishable points on an n X n X n triangular grid so that no three points are in the same row or diagonal.
%H A194478 Manuel Kauers and Christoph Koutschan, <a href="/A194478/b194478.txt">Table of n, a(n) for n = 1..1000</a> (terms 1..31 from R. H. Hardin).
%H A194478 M. Kauers and C. Koutschan, <a href="https://arxiv.org/abs/2303.02793">Some D-finite and some possibly D-finite sequences in the OEIS</a>, arXiv:2303.02793 [cs.SC], 2023.
%F A194478 From _Manuel Kauers_ and _Christoph Koutschan_, Mar 02 2023: (Start)
%F A194478 a(n) = (1/256)*(-1)^n*(2*n - 7)*(n^2 - 7*n + 13) + (1/322560)*(7*n^12 + 42*n^11 - 945*n^10 + 1274*n^9 + 26089*n^8 - 128810*n^7 + 175693*n^6 + 205366*n^5 - 810796*n^4 + 601328*n^3 + 354172*n^2 - 582180*n + 114660).
%F A194478 Recurrence: (n-2)*(14*n^11 + 70*n^10 - 2051*n^9 + 5299*n^8 + 50106*n^7 - 359946*n^6 + 953463*n^5 - 1085555*n^4 - 364412*n^3 + 3593716*n^2 - 6028304*n + 3620736)*a(n+2) + (-126*n^11 - 966*n^10 + 13377*n^9 + 4662*n^8 - 354550*n^7 + 1123664*n^6 - 1113309*n^5 + 85056*n^4 + 1719696*n^3 - 7286000*n^2 + 10210192*n - 3854400)*a(n+1) - (n+2)*(14*n^11 + 224*n^10 - 581*n^9 - 7700*n^8 + 31682*n^7 - 11948*n^6 - 91561*n^5 + 168104*n^4 - 482042*n^3 + 1253272*n^2 - 1293160*n + 383136)*a(n) = 0. (End)
%e A194478 Some solutions for 5 X 5 X 5:
%e A194478       0          0          1          0          0          1          1
%e A194478      0 0        1 0        0 0        1 1        1 0        0 0        0 1
%e A194478     1 1 0      0 0 1      0 1 0      0 0 0      1 0 1      0 1 0      1 0 0
%e A194478    0 0 1 1    0 1 0 1    0 1 1 0    1 0 0 1    0 1 0 1    0 1 1 0    0 0 1 0
%e A194478   1 0 0 0 1  1 0 0 1 0  1 0 0 0 1  0 1 1 0 0  0 1 0 0 0  0 0 1 0 1  0 0 1 1 0
%Y A194478 Column 6 of A194480.
%K A194478 nonn
%O A194478 1,5
%A A194478 _R. H. Hardin_, Aug 26 2011