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A186867 Number of 8-step king's tours on an n X n board summed over all starting positions.

%I A186867 #36 Dec 18 2023 11:31:47
%S A186867 0,0,2384,183472,1110000,3193800,6481216,10899404,16418600,23038804,
%T A186867 30760016,39582236,49505464,60529700,72654944,85881196,100208456,
%U A186867 115636724,132166000,149796284,168527576,188359876,209293184,231327500,254462824,278699156
%N A186867 Number of 8-step king's tours on an n X n board summed over all starting positions.
%C A186867 From _J. Volkmar Schmidt_, Oct 24 2023 (Start)
%C A186867 Proof of a(n) follows proof scheme from _David A. Corneth_ for A186864.
%C A186867 Distribution matrix of surrounding rectangles for 8-step walks is:
%C A186867   [0    0     0     0     0     0     0    2]
%C A186867   [0    0     0   416  3264  4224  2304  508]
%C A186867   [0    0  2384 26004 38120 26164 10080 1764]
%C A186867   [0  416 26004 67424 53320 26480  8460 1328]
%C A186867   [0 3264 38120 53320 32032 13428  3816  560]
%C A186867   [0 4224 26164 26480 13428  4952  1260  172]
%C A186867   [0 2304 10080  8460  3816  1260   288   36]
%C A186867   [2  508  1764  1328   560   172    36    4]
%C A186867 (End)
%H A186867 Andrew Howroyd, <a href="/A186867/b186867.txt">Table of n, a(n) for n = 1..1000</a>
%H A186867 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3, -3, 1).
%F A186867 Empirical: a(n) = 550504*n^2 - 3839372*n + 6382124 for n > 6.
%e A186867 Some solutions for 4 X 4:
%e A186867   0 7 6 0    2 1 0 8    0 0 1 0    0 0 6 8    3 4 5 0
%e A186867   8 0 5 1    4 3 7 0    0 0 3 2    0 0 7 5    2 0 6 0
%e A186867   0 4 3 2    0 5 6 0    0 7 5 4    2 1 4 0    1 0 7 8
%e A186867   0 0 0 0    0 0 0 0    0 8 6 0    0 3 0 0    0 0 0 0
%Y A186867 Row 8 of A186861.
%K A186867 nonn
%O A186867 1,3
%A A186867 _R. H. Hardin_, Feb 27 2011
%E A186867 a(12)-a(26) from _J. Volkmar Schmidt_, Aug 27 2023