A193983 Number of ways to arrange 5 nonattacking triangular rooks on an n X n X n triangular grid.

0, 0, 0, 0, 0, 0, 6, 270, 3195, 21273, 101484, 386052, 1243899, 3527469, 9035376, 21297492, 46838142, 97131762, 191517192, 361427508, 656353494, 1152094086, 1961910990, 3251400894, 5257953789, 8315944731, 12888836064, 19609755396

Offset

1,7

Links

R. H. Hardin, Table of n, a(n) for n = 1..52

Christopher R. H. Hanusa, Thomas Zaslavsky, A q-queens problem. VII. Combinatorial types of nonattacking chess riders, arXiv:1906.08981 [math.CO], 2019.

Formula

Empirical: a(n) = 5*a(n-1) -5*a(n-2) -14*a(n-3) +30*a(n-4) +6*a(n-5) -50*a(n-6) +10*a(n-7) +44*a(n-8) -44*a(n-10) -10*a(n-11) +50*a(n-12) -6*a(n-13) -30*a(n-14) +14*a(n-15) +5*a(n-16) -5*a(n-17) +a(n-18).

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

Empirical: G.f.: -3*x^7*(2 + 80*x + 625*x^2 + 2244*x^3 + 4898*x^4 + 7197*x^5 + 7237*x^6 + 5030*x^7 + 2294*x^8 + 633*x^9)/((-1+x)^11*(1+x)^5*(1+x+x^2)).

Empirical: a(n) = 3461*n/320 - 469*n^2/240 - 469*n^3/15 + 2383*n^4/64 - 76607*n^5/3840 + 23693*n^6/3840 - 2263*n^7/1920 + 53*n^8/384 - 7*n^9/768 + n^10/3840 + 4/3*floor(n/3) + (1359/32 - 247*n/8 + 245*n^2/32 - 13*n^3/16 + n^4/32)*floor(n/2) - 4/3*floor((1 + n)/3).

(End)

Example

Some solutions for 7 X 7 X 7
........0..............0..............0..............0..............0
.......0.0............0.0............0.0............0.0............0.0
......0.1.0..........0.1.0..........0.0.1..........1.0.0..........0.0.1
.....1.0.0.0........0.0.0.1........1.0.0.0........0.0.0.1........0.1.0.0
....0.0.0.0.1......1.0.0.0.0......0.0.0.1.0......0.1.0.0.0......1.0.0.0.0
...0.0.0.1.0.0....0.0.1.0.0.0....0.1.0.0.0.0....0.0.0.0.1.0....0.0.0.0.1.0
..0.0.1.0.0.0.0..0.0.0.0.1.0.0..0.0.0.0.1.0.0..0.0.1.0.0.0.0..0.0.0.1.0.0.0

Crossrefs

Column 5 of A193986.

Sequence in context: A244493 A221900 A229773 * A348701 A233234 A281694

Adjacent sequences: A193980 A193981 A193982 * A193984 A193985 A193986

Keyword

nonn

Author

R. H. Hardin Aug 10 2011

Status

approved