A193984 Number of ways to arrange 6 nonattacking triangular rooks on an n X n X n triangular grid.

0, 0, 0, 0, 0, 0, 0, 0, 166, 4902, 54771, 382439, 1976455, 8250687, 29309540, 91705972, 258870740, 671005444, 1618468198, 3670491998, 7891420850, 16191666766, 31878943876, 60501909500, 111108316262, 198083991586, 343784805209

Offset: 1

Author

R. H. Hardin Aug 10 2011

Keywords

• nonn

Examples

			Some solutions for 9 X 9 X 9
..........0..................0..................0..................0
.........0.0................0.0................0.0................0.1
........1.0.0..............0.0.0..............0.0.1..............0.0.0
.......0.0.0.0............1.0.0.0............0.0.0.0............1.0.0.0
......0.0.0.1.0..........0.0.0.1.0..........1.0.0.0.0..........0.0.0.0.0
.....0.0.1.0.0.0........0.1.0.0.0.0........0.0.0.1.0.0........0.0.0.1.0.0
....0.0.0.0.0.0.1......0.0.0.0.0.0.1......0.1.0.0.0.0.0......0.0.0.0.0.1.0
...0.1.0.0.0.0.0.0....0.0.0.0.0.1.0.0....0.0.0.0.1.0.0.0....0.0.1.0.0.0.0.0
..0.0.0.0.1.0.0.0.0..0.0.1.0.0.0.0.0.0..0.0.0.0.0.0.0.1.0..0.0.0.0.1.0.0.0.0
		

Links

• R. H. Hardin, Table of n, a(n) for n = 1..37
• Christopher R. H. Hanusa, Thomas Zaslavsky, A q-queens problem. VII. Combinatorial types of nonattacking chess riders, arXiv:1906.08981 [math.CO], 2019.

Crossrefs

Column 6 of A193986.

Formula

Contribution from Vaclav Kotesovec, Aug 31 2012: (Start)

Empirical: Recurrence: a(n-29) - 4*a(n-28) + 17*a(n-26) - 9*a(n-25) - 32*a(n-24) + 7*a(n-23) + 51*a(n-22) + 26*a(n-21) - 77*a(n-20) - 59*a(n-19) + 58*a(n-18) + 74*a(n-17) + 21*a(n-16) - 74*a(n-15) - 74*a(n-14) + 21*a(n-13) + 74*a(n-12) + 58*a(n-11) - 59*a(n-10) - 77*a(n-9) + 26*a(n-8) + 51*a(n-7) + 7*a(n-6) - 32*a(n-5) - 9*a(n-4) + 17*a(n-3) - 4*a(n-1) + a(n) = 0.

Empirical: G.f.: -x^9*(166 + 4404*x + 39567*x^2 + 205744*x^3 + 734283*x^4 + 1960827*x^5 + 4120441*x^6 + 7036145*x^7 + 9956248*x^8 + 11823233*x^9 + 11839707*x^10 + 10002936*x^11 + 7077533*x^12 + 4145811*x^13 + 1957821*x^14 + 721991*x^15 + 191674*x^16 + 31709*x^17)/((-1+x)^13*(1+x)^7*(1+x^2)*(1+x+x^2)^3).

Empirical: a(n) = 98227*n/10080 + 39907*n^2/180 - 1105267*n^3/1920 + 6516731*n^4/11520 - 7025857*n^5/23040 + 4788163*n^6/46080 - 3842803*n^7/161280 + 34619*n^8/9216 - 9299*n^9/23040 + 29*n^10/1024 - 3*n^11/2560 + n^12/46080 + 25/4*floor(n/4) + (56 - 38*n/3 + 2*n^2/3)*floor(n/3) + (12053/16 - 32515*n/48 + 22687*n^2/96 - 8249*n^3/192 + 279*n^4/64 - 15*n^5/64 + n^6/192)*floor(n/2) - 3*floor((1+n)/4) + (-184/3 + 38*n/3 - 2*n^2/3)*floor((1+n)/3).

(End)