A194493 Number of ways to arrange 3 nonattacking queens on the lower triangle of an n X n board.

0, 0, 0, 0, 12, 82, 330, 1008, 2566, 5742, 11652, 21926, 38802, 65322, 105428, 164214, 248022, 364764, 523998, 737334, 1018488, 1383768, 1852104, 2445628, 3189660, 4113396, 5249848, 6636636, 8315880, 10335110, 12747090, 15610860, 18991490

Offset: 1

Author

R. H. Hardin Aug 26 2011

Keywords

• nonn

Comments

Column 3 of A194498

Examples

			Some solutions for 5X5
..0..........0..........0..........0..........0..........0..........1
..1.0........0.1........1.0........0.0........1.0........0.1........0.0
..0.0.0......0.0.0......0.0.0......1.0.0......0.0.1......0.0.0......0.1.0
..0.1.0.0....1.0.0.0....0.0.0.1....0.0.0.1....0.0.0.0....1.0.0.0....0.0.0.0
..0.0.0.0.1..0.0.1.0.0..0.1.0.0.0..0.1.0.0.0..0.1.0.0.0..0.0.0.1.0..0.0.1.0.0
		

Links

• R. H. Hardin, Table of n, a(n) for n = 1..200

Formula

Empirical: a(n) = 3*a(n-1) -7*a(n-3) +3*a(n-4) +6*a(n-5) -6*a(n-7) -3*a(n-8) +7*a(n-9) -3*a(n-11) +a(n-12), [R. H. Hardin Aug 26 2011]

G.f.: -2*x^5*(18*x^5 + 40*x^4 + 51*x^3 + 42*x^2 + 23*x + 6)/((x-1)^7*(x+1)^3*(x^2+x+1))

Explicit formula: n^6/48 - 11*n^5/48 + 15*n^4/16 - 241*n^3/144 + 17*n^2/16 - 17*n/144 + (n^2/8 - 9*n/8 + 17/8)*floor(n/2) + 2/3*floor(n/3), [Vaclav Kotesovec, Apr 08 2012]