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.

0, 0, 0, 1, 337, 8733, 96478, 668028, 3413828, 14054915, 49171641, 151422970, 420674150, 1073422309, 2550004472, 5699074284, 12082541388, 24462528078, 47555986746, 89173692795, 161899772067, 285517344145, 490447009030

Offset: 1

Author

R. H. Hardin, Aug 26 2011

Keywords

• nonn

Examples

			Some solutions for 5 X 5 X 5:
      0          0          1          0          0          1          1
     0 0        1 0        0 0        1 1        1 0        0 0        0 1
    1 1 0      0 0 1      0 1 0      0 0 0      1 0 1      0 1 0      1 0 0
   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
  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
		

Links

• Manuel Kauers and Christoph Koutschan, Table of n, a(n) for n = 1..1000 (terms 1..31 from R. H. Hardin).
• M. Kauers and C. Koutschan, Some D-finite and some possibly D-finite sequences in the OEIS, arXiv:2303.02793 [cs.SC], 2023.

Crossrefs

Column 6 of A194480.

Formula

From Manuel Kauers and Christoph Koutschan, Mar 02 2023: (Start)

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).

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)