A187155 T(n,k)=Number of n-step one space at a time bishop's tours on a kXk board summed over all starting positions.

1, 4, 0, 9, 4, 0, 16, 16, 0, 0, 25, 36, 20, 0, 0, 36, 64, 64, 8, 0, 0, 49, 100, 132, 92, 0, 0, 0, 64, 144, 224, 248, 72, 0, 0, 0, 81, 196, 340, 476, 388, 56, 0, 0, 0, 100, 256, 480, 776, 904, 456, 16, 0, 0, 0, 121, 324, 644, 1148, 1620, 1588, 544, 0, 0, 0, 0, 144, 400, 832, 1592

Offset: 1

Author

R. H. Hardin, Mar 06 2011

Keywords

• nonn
• tabl

Examples

			Table starts
.1.4..9.16..25...36....49....64.....81....100....121....144....169.....196
.0.4.16.36..64..100...144...196....256....324....400....484....576.....676
.0.0.20.64.132..224...340...480....644....832...1044...1280...1540....1824
.0.0..8.92.248..476...776..1148...1592...2108...2696...3356...4088....4892
.0.0..0.72.388..904..1620..2536...3652...4968...6484...8200..10116...12232
.0.0..0.56.456.1588..3288..5556...8392..11796..15768..20308..25416...31092
.0.0..0.16.544.2328..6172.11576..18540..27064..37148..48792..61996...76760
.0.0..0..0.472.3504.10576.23340..40448..61900..87696.117836.152320..191148
.0.0..0..0.392.4216.17696.43136..83844.136384.200756.276960.364996..464864
.0.0..0..0.168.5472.26912.80392.168104.296708.457848.651524.877736.1136484
Some n=4 solutions for 4X4
..0..0..0..1....0..0..1..0....4..0..0..0....0..4..0..0....0..0..1..0
..0..0..2..0....0..2..0..0....0..3..0..0....3..0..1..0....0..2..0..4
..0..3..0..0....0..0..3..0....0..0..2..0....0..2..0..0....0..0..3..0
..0..0..4..0....0..4..0..0....0..0..0..1....0..0..0..0....0..0..0..0
		

Links

• R. H. Hardin, Table of n, a(n) for n = 1..287

Crossrefs

Row 2 is A016742(n-1)

Formula

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

Empirical: T(2,k) = 4*k^2 - 8*k + 4

Empirical: T(3,k) = 12*k^2 - 40*k + 32 for k>1

Empirical: T(4,k) = 36*k^2 - 168*k + 188 for k>2

Empirical: T(5,k) = 100*k^2 - 584*k + 808 for k>3

Empirical: T(6,k) = 284*k^2 - 1992*k + 3316 for k>4

Empirical: T(7,k) = 780*k^2 - 6296*k + 12024 for k>5

Empirical: T(8,k) = 2172*k^2 - 19816*k + 42860 for k>6