A194494 Number of ways to arrange 4 nonattacking queens on the lower triangle of an n X n board.

0, 0, 0, 0, 0, 8, 118, 802, 3708, 13280, 39734, 104000, 244948, 530632, 1072776, 2048056, 3723314, 6492342, 10915254, 17777372, 28147380, 43465356, 65624634, 97098802, 141050688, 201509798, 283514112, 393348562, 538725268, 729098516

Offset: 1

Author

R. H. Hardin Aug 26 2011

Keywords

• nonn

Comments

Column 4 of A194498

Examples

			Some solutions for 6X6
..0............0............0............0............0............0
..1.0..........0.0..........0.0..........0.1..........0.0..........0.1
..0.0.0........1.0.0........0.0.1........0.0.0........0.1.0........0.0.0
..0.1.0.0......0.0.1.0......1.0.0.0......1.0.0.0......0.0.0.1......0.0.1.0
..0.0.0.0.1....0.0.0.0.1....0.0.0.1.0....0.0.1.0.0....1.0.0.0.0....1.0.0.0.0
..0.0.1.0.0.0..0.1.0.0.0.0..0.1.0.0.0.0..0.0.0.0.1.0..0.0.1.0.0.0..0.0.0.1.0.0
		

Links

• R. H. Hardin, Table of n, a(n) for n = 1..81

Formula

Empirical: a(n) = 4*a(n-2) +3*a(n-3) -5*a(n-4) -11*a(n-5) -3*a(n-6) +11*a(n-7) +14*a(n-8) +6*a(n-9) -7*a(n-10) -16*a(n-11) -14*a(n-12) +14*a(n-14) +16*a(n-15) +7*a(n-16) -6*a(n-17) -14*a(n-18) -11*a(n-19) +3*a(n-20) +11*a(n-21) +5*a(n-22) -3*a(n-23) -4*a(n-24) +a(n-26), [R. H. Hardin, Aug 26 2011]

G.f.: -2*x^6*(287*x^18 + 1545*x^17 + 4929*x^16 + 11689*x^15 + 22673*x^14 + 36995*x^13 + 51875*x^12 + 63203*x^11 + 67465*x^10 + 63168*x^9 + 51807*x^8 + 36900*x^7 + 22544*x^6 + 11587*x^5 + 4879*x^4 + 1606*x^3 + 385*x^2 + 59*x + 4)/((x-1)^9*(x+1)^5*(x^2+1)*(x^2+x+1)^3*(x^4+x^3+x^2+x+1))

Explicit formula: n^8/384 - n^7/16 + 23*n^6/36 - 1301*n^5/360 + 14125*n^4/1152 - 8013*n^3/320 + 83147*n^2/2880 - 2089*n/160 + (n^4/32 - 17*n^3/24 + 193*n^2/32 - 2209*n/96 + 439/16)*floor(n/2) + (n^2/3 - 13*n/3 + 136/9)*floor(n/3) - 28/9*floor((n+1)/3) + 23/4*floor(n/4) - 3*floor((n+1)/4) + 4/5*floor(n/5) - 2/5*floor((n+2)/5) - 2/5*floor((n+3)/5), [Vaclav Kotesovec, Apr 08 2012]