A187507 T(n,k)=Number of n-step S, E, and NW-moving king's tours on a kXk board summed over all starting positions

1, 4, 0, 9, 5, 0, 16, 16, 6, 0, 25, 33, 31, 2, 0, 36, 56, 74, 36, 0, 0, 49, 85, 135, 115, 40, 0, 0, 64, 120, 214, 236, 184, 36, 0, 0, 81, 161, 311, 399, 435, 272, 20, 0, 0, 100, 208, 426, 604, 788, 772, 330, 12, 0, 0, 121, 261, 559, 851, 1243, 1525, 1224, 390, 6, 0, 0, 144, 320, 710

Offset

1,2

Comments

Table starts

.1.4..9..16...25....36....49....64....81....100....121....144....169....196

.0.5.16..33...56....85...120...161...208....261....320....385....456....533

.0.6.31..74..135...214...311...426...559....710....879...1066...1271...1494

.0.2.36.115..236...399...604...851..1140...1471...1844...2259...2716...3215

.0.0.40.184..435...788..1243..1800..2459...3220...4083...5048...6115...7284

.0.0.36.272..772..1525..2524..3769..5260...6997...8980..11209..13684..16405

.0.0.20.330.1224..2726..4807..7458.10679..14470..18831..23762..29263..35334

.0.0.12.390.1910..4880..9250.14969.22026..30421..40154..51225..63634..77381

.0.0..6.450.2872..8522.17564.29834.45255..63814..85511.110346.138319.169430

.0.0..0.398.3868.13796.31548.56952.89684.129637.176796.231161.292732.361509

Links

R. H. Hardin, Table of n, a(n) for n = 1..290

Formula

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

Empirical: T(2,k) = 3*k^2 - 4*k + 1

Empirical: T(3,k) = 9*k^2 - 20*k + 10 for k>1

Empirical: T(4,k) = 21*k^2 - 68*k + 51 for k>2

Empirical: T(5,k) = 51*k^2 - 208*k + 200 for k>3

Empirical: T(6,k) = 123*k^2 - 600*k + 697 for k>4

Empirical: T(7,k) = 285*k^2 - 1624*k + 2210 for k>5

Empirical: T(8,k) = 669*k^2 - 4316*k + 6681 for k>6

Empirical: T(9,k) = 1569*k^2 - 11252*k + 19434 for k>7

Empirical: T(10,k) = 3603*k^2 - 28504*k + 54377 for k>8

Example

Some n=4 solutions for 4X4
..0..0..0..0....0..0..0..0....1..0..0..0....0..0..0..0....0..0..0..0
..0..0..0..0....0..0..0..0....2..0..0..0....0..0..0..0....2..3..4..0
..0..4..2..0....0..3..4..0....3..0..0..0....3..1..0..0....0..1..0..0
..0..0..3..1....0..1..2..0....4..0..0..0....4..2..0..0....0..0..0..0

Crossrefs

Row 2 is A045944(n-1)

Sequence in context: A330422 A035102 A242015 * A187857 A215499 A190262

Adjacent sequences: A187504 A187505 A187506 * A187508 A187509 A187510

Keyword

nonn,tabl

Author

R. H. Hardin Mar 10 2011

Status

approved