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

1,9

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.

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)

Example

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

Crossrefs

Column 6 of A193986.

Sequence in context: A204971 A204964 A241969 * A144380 A204963 A011815

Adjacent sequences: A193981 A193982 A193983 * A193985 A193986 A193987

Keyword

nonn

Author

R. H. Hardin Aug 10 2011

Status

approved