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