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

Author

R. H. Hardin Mar 11 2011

Keywords

• nonn
• tabl

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

Examples

			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
		

Links

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

Crossrefs

Row 2 is A002943(n-1)

Row 3 is A152750(n-1)

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