A187586 T(n,k)=Number of n-step E, S, NW and NE-moving king's tours on a kXk board summed over all starting positions

1, 4, 0, 9, 6, 0, 16, 20, 8, 0, 25, 42, 48, 5, 0, 36, 72, 120, 84, 0, 0, 49, 110, 224, 286, 106, 0, 0, 64, 156, 360, 604, 578, 104, 0, 0, 81, 210, 528, 1038, 1484, 1069, 78, 0, 0, 100, 272, 728, 1588, 2794, 3514, 1708, 34, 0, 0, 121, 342, 960, 2254, 4508, 7480, 7666, 2309, 13

Offset

1,2

Comments

Table starts

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

.0.6..20...42....72....110....156.....210.....272.....342.....420.....506

.0.8..48..120...224....360....528.....728.....960....1224....1520....1848

.0.5..84..286...604...1038...1588....2254....3036....3934....4948....6078

.0.0.106..578..1484...2794...4508....6626....9148...12074...15404...19138

.0.0.104.1069..3514...7480..12874...19696...27946...37624...48730...61264

.0.0..78.1708..7666..19104..35832...57592...84384..116208..153064..194952

.0.0..34.2309.15056..45718..95776..164135..250132..353767..475040..613951

.0.0..13.2792.27252.103108.246792..458018..732810.1069534.1468190.1928778

.0.0...0.3108.45960.219432.609070.1243461.2111652.3201436.4508924

Links

R. H. Hardin, Table of n, a(n) for n = 1..200

Formula

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

Empirical: T(2,k) = 4*k^2 - 6*k + 2

Empirical: T(3,k) = 16*k^2 - 40*k + 24

Empirical: T(4,k) = 58*k^2 - 204*k + 174 for k>2

Empirical: T(5,k) = 202*k^2 - 912*k + 994 for k>3

Empirical: T(6,k) = 714*k^2 - 3888*k + 5104 for k>4

Empirical: T(7,k) = 2516*k^2 - 15980*k + 24408 for k>5

Empirical: T(8,k) = 8819*k^2 - 63926*k + 111127 for k>6

Empirical: T(9,k) = 30966*k^2 - 251630*k + 489234 for k>7

Empirical: T(10,k) = 108852*k^2 - 978404*k + 2100276 for k>8

Example

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

Crossrefs

Row 2 is A002943(n-1)

Row 3 is A152750(n-1)

Sequence in context: A187857 A215499 A190262 * A198866 A269720 A245638

Adjacent sequences: A187583 A187584 A187585 * A187587 A187588 A187589

Keyword

nonn,tabl

Author

R. H. Hardin Mar 11 2011

Status

approved