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A187172 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.

%I A187172 #11 Oct 27 2019 20:58:46
%S A187172 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 A187172 120,128,48,0,0,0,0,81,168,220,176,16,0,0,0,0,100,224,336,384,136,0,0,
%U A187172 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 A187172 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 A187172 Table starts
%C A187172 1 4 9 16 25  36  49   64   81   100   121    144    169    196    225    256
%C A187172 0 0 8 24 48  80 120  168  224   288   360    440    528    624    728    840
%C A187172 0 0 0 16 60 128 220  336  476   640   828   1040   1276   1536   1820   2128
%C A187172 0 0 0  8 48 176 384  664 1016  1440  1936   2504   3144   3856   4640   5496
%C A187172 0 0 0  0 16 136 456 1024 1804  2784  3964   5344   6924   8704  10684  12864
%C A187172 0 0 0  0  0  88 496 1440 3064  5344  8208  11640  15640  20208  25344  31048
%C A187172 0 0 0  0  0  16 368 1600 4284  8760 15104  23144  32764  43944  56684  70984
%C A187172 0 0 0  0  0   0 280 1784 5944 14072 27104  45288  68400  96048 128064 164424
%C A187172 0 0 0  0  0   0  88 1440 6828 19840 43668  80624 131576 196192 273592 363080
%C A187172 0 0 0  0  0   0   8 1088 7896 27984 70344 142816 250728 396808 580696 800584
%H A187172 R. H. Hardin, <a href="/A187172/b187172.txt">Table of n, a(n) for n = 1..310</a>
%F A187172 Empirical:
%F A187172   T(1,k) =      k^2;
%F A187172   T(2,k) =    4*k^2 -    12*k +      8;
%F A187172   T(3,k) =   12*k^2 -    64*k +     80 for k >  3;
%F A187172   T(4,k) =   36*k^2 -   260*k +    440 for k >  5;
%F A187172   T(5,k) =  100*k^2 -   920*k +   1984 for k >  7;
%F A187172   T(6,k) =  284*k^2 -  3100*k +   7944 for k >  9;
%F A187172   T(7,k) =  780*k^2 -  9880*k +  29384 for k > 11;
%F A187172   T(8,k) = 2172*k^2 - 30972*k + 103944 for k > 13.
%e A187172 One of 98568 n=51 solutions for 16 X 16:
%e A187172    0  1  0  0  0  0  4  0  0  0  0  7  0  0  0  0
%e A187172    0  0  0  0  3  0  0  0  0  6  0  0  0  0  9  0
%e A187172    0  0  2  0  0  0  0  5  0  0  0  0  8  0  0  0
%e A187172   51  0  0  0  0 16  0  0  0  0 13  0  0  0  0 10
%e A187172    0  0  0 17  0  0  0  0 14  0  0  0  0 11  0  0
%e A187172    0 50  0  0  0  0 15  0  0  0  0 12  0  0  0  0
%e A187172    0  0  0  0 18  0  0  0  0 21  0  0  0  0 24  0
%e A187172    0  0 49  0  0  0  0 20  0  0  0  0 23  0  0  0
%e A187172   48  0  0  0  0 19  0  0  0  0 22  0  0  0  0 25
%e A187172    0  0  0 46  0  0  0  0 35  0  0  0  0 26  0  0
%e A187172    0 47  0  0  0  0 36  0  0  0  0 27  0  0  0  0
%e A187172    0  0  0  0 45  0  0  0  0 34  0  0  0  0 29  0
%e A187172    0  0 44  0  0  0  0 37  0  0  0  0 28  0  0  0
%e A187172   43  0  0  0  0 40  0  0  0  0 33  0  0  0  0 30
%e A187172    0  0  0 41  0  0  0  0 38  0  0  0  0 31  0  0
%e A187172    0 42  0  0  0  0 39  0  0  0  0 32  0  0  0  0
%Y A187172 Row 2 is A033996(n-2).
%K A187172 nonn,tabl
%O A187172 1,2
%A A187172 _R. H. Hardin_, Mar 06 2011