A188777 T(n,k) = Number of n-turn bishop's tours on a k X k board summed over all starting positions.

1, 4, 0, 9, 4, 0, 16, 20, 0, 0, 25, 56, 28, 0, 0, 36, 120, 152, 24, 0, 0, 49, 220, 488, 328, 8, 0, 0, 64, 364, 1192, 1720, 584, 0, 0, 0, 81, 560, 2468, 5816, 5464, 840, 0, 0, 0, 100, 816, 4560, 15424, 26360, 15824, 784, 0, 0, 0, 121, 1140, 7760, 34736, 91120, 112680, 40496

Offset: 1

Author

R. H. Hardin, Apr 10 2011

Keywords

• nonn
• tabl

Comments

Table starts

.1.4..9..16.....25......36.......49.......64.......81......100.....121....144

.0.4.20..56....120.....220......364......560......816.....1140....1540...2024

.0.0.28.152....488....1192.....2468.....4560.....7760....12400...18860..27560

.0.0.24.328...1720....5816....15424....34736....69776...128528..221448.361528

.0.0..8.584...5464...26360....91120...252720...603696..1288592.2525400

.0.0..0.840..15824..112680...516160..1778608..5082912.12622640

.0.0..0.784..40496..451104..2803552.12139552.41792672

.0.0..0.384..88264.1665344.14497784.80088992

.0.0..0...0.159704.5607456

.0.0..0...0.229296

Examples

			Some n=4 solutions for 4 X 4
..0..0..0..0....0..4..0..1....0..0..0..4....3..0..0..0....0..1..0..0
..0..3..0..0....0..0..3..0....0..0..0..0....0..2..0..0....4..0..2..0
..2..0..4..0....0..2..0..0....0..3..0..1....0..0..4..0....0..3..0..0
..0..1..0..0....0..0..0..0....0..0..2..0....0..0..0..1....0..0..0..0
		

Links

• R. H. Hardin, Table of n, a(n) for n = 1..113

Crossrefs

Row 2 is A002492(n-1).

Formula

Empirical: T(1,k) = k^2.

Empirical: T(2,k) = (4/3)*k^3 - 2*k^2 + (2/3)*k.

Empirical: T(3,k) = 4*T(3,k-1)-5*T(3,k-2)+5*T(3,k-4)-4*T(3,k-5)+T(3,k-6).

Empirical: T(4,k) = 4*T(4,k-1)-4*T(4,k-2)-4*T(4,k-3)+10*T(4,k-4)-4*T(4,k-5)-4*T(4,k-6)+4*T(4,k-7)-T(4,k-8).

Empirical: T(5,k) = 4*T(5,k-1)-3*T(5,k-2)-8*T(5,k-3)+14*T(5,k-4)-14*T(5,k-6)+8*T(5,k-7)+3*T(5,k-8)-4*T(5,k-9)+T(5,k-10).

Empirical: T(6,k) = 4*T(6,k-1)-2*T(6,k-2)-12*T(6,k-3)+17*T(6,k-4)+8*T(6,k-5)-28*T(6,k-6)+8*T(6,k-7)+17*T(6,k-8)-12*T(6,k-9)-2*T(6,k-10)+4*T(6,k-11)-T(6,k-12).