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%I #11 Oct 27 2019 20:58:46
%S 1,4,0,9,0,0,16,8,0,0,25,24,0,0,0,36,48,16,0,0,0,49,80,60,8,0,0,0,64,
%T 120,128,48,0,0,0,0,81,168,220,176,16,0,0,0,0,100,224,336,384,136,0,0,
%U 0,0,0,121,288,476,664,456,88,0,0,0,0,0,144,360,640,1016,1024,496,16,0,0,0,0,0
%N T(n,k) is the number of n-step left-handed knight's tours (moves only out two, left one) on a k X k board summed over all starting positions.
%C Table starts
%C 1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 256
%C 0 0 8 24 48 80 120 168 224 288 360 440 528 624 728 840
%C 0 0 0 16 60 128 220 336 476 640 828 1040 1276 1536 1820 2128
%C 0 0 0 8 48 176 384 664 1016 1440 1936 2504 3144 3856 4640 5496
%C 0 0 0 0 16 136 456 1024 1804 2784 3964 5344 6924 8704 10684 12864
%C 0 0 0 0 0 88 496 1440 3064 5344 8208 11640 15640 20208 25344 31048
%C 0 0 0 0 0 16 368 1600 4284 8760 15104 23144 32764 43944 56684 70984
%C 0 0 0 0 0 0 280 1784 5944 14072 27104 45288 68400 96048 128064 164424
%C 0 0 0 0 0 0 88 1440 6828 19840 43668 80624 131576 196192 273592 363080
%C 0 0 0 0 0 0 8 1088 7896 27984 70344 142816 250728 396808 580696 800584
%H R. H. Hardin, <a href="/A187172/b187172.txt">Table of n, a(n) for n = 1..310</a>
%F Empirical:
%F T(1,k) = k^2;
%F T(2,k) = 4*k^2 - 12*k + 8;
%F T(3,k) = 12*k^2 - 64*k + 80 for k > 3;
%F T(4,k) = 36*k^2 - 260*k + 440 for k > 5;
%F T(5,k) = 100*k^2 - 920*k + 1984 for k > 7;
%F T(6,k) = 284*k^2 - 3100*k + 7944 for k > 9;
%F T(7,k) = 780*k^2 - 9880*k + 29384 for k > 11;
%F T(8,k) = 2172*k^2 - 30972*k + 103944 for k > 13.
%e One of 98568 n=51 solutions for 16 X 16:
%e 0 1 0 0 0 0 4 0 0 0 0 7 0 0 0 0
%e 0 0 0 0 3 0 0 0 0 6 0 0 0 0 9 0
%e 0 0 2 0 0 0 0 5 0 0 0 0 8 0 0 0
%e 51 0 0 0 0 16 0 0 0 0 13 0 0 0 0 10
%e 0 0 0 17 0 0 0 0 14 0 0 0 0 11 0 0
%e 0 50 0 0 0 0 15 0 0 0 0 12 0 0 0 0
%e 0 0 0 0 18 0 0 0 0 21 0 0 0 0 24 0
%e 0 0 49 0 0 0 0 20 0 0 0 0 23 0 0 0
%e 48 0 0 0 0 19 0 0 0 0 22 0 0 0 0 25
%e 0 0 0 46 0 0 0 0 35 0 0 0 0 26 0 0
%e 0 47 0 0 0 0 36 0 0 0 0 27 0 0 0 0
%e 0 0 0 0 45 0 0 0 0 34 0 0 0 0 29 0
%e 0 0 44 0 0 0 0 37 0 0 0 0 28 0 0 0
%e 43 0 0 0 0 40 0 0 0 0 33 0 0 0 0 30
%e 0 0 0 41 0 0 0 0 38 0 0 0 0 31 0 0
%e 0 42 0 0 0 0 39 0 0 0 0 32 0 0 0 0
%Y Row 2 is A033996(n-2).
%K nonn,tabl
%O 1,2
%A _R. H. Hardin_, Mar 06 2011
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