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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.

%I A187586 #8 Jul 22 2025 10:49:01
%S A187586 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,
%T A187586 106,0,0,64,156,360,604,578,104,0,0,81,210,528,1038,1484,1069,78,0,0,
%U A187586 100,272,728,1588,2794,3514,1708,34,0,0,121,342,960,2254,4508,7480,7666,2309,13
%N 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.
%C A187586 Table starts
%C A187586 .1.4...9...16....25.....36.....49......64......81.....100.....121.....144
%C A187586 .0.6..20...42....72....110....156.....210.....272.....342.....420.....506
%C A187586 .0.8..48..120...224....360....528.....728.....960....1224....1520....1848
%C A187586 .0.5..84..286...604...1038...1588....2254....3036....3934....4948....6078
%C A187586 .0.0.106..578..1484...2794...4508....6626....9148...12074...15404...19138
%C A187586 .0.0.104.1069..3514...7480..12874...19696...27946...37624...48730...61264
%C A187586 .0.0..78.1708..7666..19104..35832...57592...84384..116208..153064..194952
%C A187586 .0.0..34.2309.15056..45718..95776..164135..250132..353767..475040..613951
%C A187586 .0.0..13.2792.27252.103108.246792..458018..732810.1069534.1468190.1928778
%C A187586 .0.0...0.3108.45960.219432.609070.1243461.2111652.3201436.4508924
%H A187586 R. H. Hardin, <a href="/A187586/b187586.txt">Table of n, a(n) for n = 1..200</a>
%F A187586 Empirical: T(1,k) = k^2
%F A187586 Empirical: T(2,k) = 4*k^2 - 6*k + 2
%F A187586 Empirical: T(3,k) = 16*k^2 - 40*k + 24
%F A187586 Empirical: T(4,k) = 58*k^2 - 204*k + 174 for k>2
%F A187586 Empirical: T(5,k) = 202*k^2 - 912*k + 994 for k>3
%F A187586 Empirical: T(6,k) = 714*k^2 - 3888*k + 5104 for k>4
%F A187586 Empirical: T(7,k) = 2516*k^2 - 15980*k + 24408 for k>5
%F A187586 Empirical: T(8,k) = 8819*k^2 - 63926*k + 111127 for k>6
%F A187586 Empirical: T(9,k) = 30966*k^2 - 251630*k + 489234 for k>7
%F A187586 Empirical: T(10,k) = 108852*k^2 - 978404*k + 2100276 for k>8
%e A187586 Some k=4 solutions for 4X4
%e A187586 ..0..0..0..0....3..0..0..0....0..0..0..0....0..4..0..0....0..0..0..3
%e A187586 ..0..0..0..0....4..2..0..0....4..2..0..0....3..0..0..0....0..0..2..4
%e A187586 ..0..1..2..0....1..0..0..0....0..3..1..0....0..2..0..0....0..0..0..1
%e A187586 ..0..0..3..4....0..0..0..0....0..0..0..0....1..0..0..0....0..0..0..0
%Y A187586 Row 2 is A002943(n-1)
%Y A187586 Row 3 is A152750(n-1)
%K A187586 nonn,tabl
%O A187586 1,2
%A A187586 _R. H. Hardin_ Mar 11 2011