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

1,2

Links

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

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

Example

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

Crossrefs

Row 2 is A016742(n-1)

Sequence in context: A198238 A344716 A187377 * A187296 A187046 A188777

Adjacent sequences: A187152 A187153 A187154 * A187156 A187157 A187158

Keyword

nonn,tabl

Author

R. H. Hardin, Mar 06 2011

Status

approved