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A366829 Number of 9-step self-avoiding king's tours on an n X n board summed over all starting positions.

%I A366829 #28 Dec 17 2023 10:28:33
%S A366829 0,0,784,436984,3908376,13530576,30543072,54738536,85743256,123447704,
%T A366829 167851880,218955784,276759416,341262776,412465864,490368680,
%U A366829 574971224,666273496,764275496,868977224,980378680,1098479864,1223280776,1354781416,1492981784,1637881880
%N A366829 Number of 9-step self-avoiding king's tours on an n X n board summed over all starting positions.
%C A366829 Proof of the formula follows proof scheme from _David A. Corneth_ for A186864.
%C A366829 Distribution matrix of surrounding rectangles for 9-step walks is:
%C A366829   [0     0      0      0      0      0     0     0    2]
%C A366829   [0     0      0      0   3584  10496 10752  5120 1020]
%C A366829   [0     0    784  43856 129100 136320 83208 29160 4680]
%C A366829   [0     0  43856 258424 318816 215096 99984 29680 4296]
%C A366829   [0  3584 129100 318816 262816 142888 57376 15400 2100]
%C A366829   [0 10496 136320 215096 142888  67688 24288  5960  768]
%C A366829   [0 10752  83208  99984  57376  24288  7864  1760  212]
%C A366829   [0  5120  29160  29680  15400   5960  1760   360   40]
%C A366829   [2  1020   4680   4296   2100    768   212    40    4]
%H A366829 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A366829 a(n) = 3349864*n^2 - 25942968*n + 47890984 for n>7.
%F A366829 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 10. - _Stefano Spezia_, Oct 28 2023
%e A366829 Some solutions for 3 X 3:
%e A366829   1 2 3  1 2 3  1 2 3  1 2 3  1 7 8  1 2 8
%e A366829   4 5 6  6 5 4  8 9 4  7 6 4  6 2 9  3 7 9
%e A366829   7 8 9  7 8 9  7 6 5  8 9 5  5 4 3  4 5 6
%Y A366829 Row 9 of A186861.
%K A366829 nonn,easy,walk
%O A366829 1,3
%A A366829 _J. Volkmar Schmidt_, Oct 25 2023