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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.

%I A187507 #8 Jul 22 2025 10:46:52
%S A187507 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,
%T A187507 0,0,64,120,214,236,184,36,0,0,81,161,311,399,435,272,20,0,0,100,208,
%U A187507 426,604,788,772,330,12,0,0,121,261,559,851,1243,1525,1224,390,6,0,0,144,320,710
%N 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.
%C A187507 Table starts
%C A187507 .1.4..9..16...25....36....49....64....81....100....121....144....169....196
%C A187507 .0.5.16..33...56....85...120...161...208....261....320....385....456....533
%C A187507 .0.6.31..74..135...214...311...426...559....710....879...1066...1271...1494
%C A187507 .0.2.36.115..236...399...604...851..1140...1471...1844...2259...2716...3215
%C A187507 .0.0.40.184..435...788..1243..1800..2459...3220...4083...5048...6115...7284
%C A187507 .0.0.36.272..772..1525..2524..3769..5260...6997...8980..11209..13684..16405
%C A187507 .0.0.20.330.1224..2726..4807..7458.10679..14470..18831..23762..29263..35334
%C A187507 .0.0.12.390.1910..4880..9250.14969.22026..30421..40154..51225..63634..77381
%C A187507 .0.0..6.450.2872..8522.17564.29834.45255..63814..85511.110346.138319.169430
%C A187507 .0.0..0.398.3868.13796.31548.56952.89684.129637.176796.231161.292732.361509
%H A187507 R. H. Hardin, <a href="/A187507/b187507.txt">Table of n, a(n) for n = 1..290</a>
%F A187507 Empirical: T(1,k) = k^2
%F A187507 Empirical: T(2,k) = 3*k^2 - 4*k + 1
%F A187507 Empirical: T(3,k) = 9*k^2 - 20*k + 10 for k>1
%F A187507 Empirical: T(4,k) = 21*k^2 - 68*k + 51 for k>2
%F A187507 Empirical: T(5,k) = 51*k^2 - 208*k + 200 for k>3
%F A187507 Empirical: T(6,k) = 123*k^2 - 600*k + 697 for k>4
%F A187507 Empirical: T(7,k) = 285*k^2 - 1624*k + 2210 for k>5
%F A187507 Empirical: T(8,k) = 669*k^2 - 4316*k + 6681 for k>6
%F A187507 Empirical: T(9,k) = 1569*k^2 - 11252*k + 19434 for k>7
%F A187507 Empirical: T(10,k) = 3603*k^2 - 28504*k + 54377 for k>8
%e A187507 Some n=4 solutions for 4X4
%e A187507 ..0..0..0..0....0..0..0..0....1..0..0..0....0..0..0..0....0..0..0..0
%e A187507 ..0..0..0..0....0..0..0..0....2..0..0..0....0..0..0..0....2..3..4..0
%e A187507 ..0..4..2..0....0..3..4..0....3..0..0..0....3..1..0..0....0..1..0..0
%e A187507 ..0..0..3..1....0..1..2..0....4..0..0..0....4..2..0..0....0..0..0..0
%Y A187507 Row 2 is A045944(n-1)
%K A187507 nonn,tabl
%O A187507 1,2
%A A187507 _R. H. Hardin_ Mar 10 2011