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A186863 Number of 4-step king's tours on an n X n board summed over all starting positions.

%I A186863 #58 Dec 18 2023 11:26:30
%S A186863 0,24,496,1764,3768,6508,9984,14196,19144,24828,31248,38404,46296,
%T A186863 54924,64288,74388,85224,96796,109104,122148,135928,150444,165696,
%U A186863 181684,198408,215868,234064,252996,272664,293068,314208,336084,358696,382044,406128,430948,456504,482796,509824
%N A186863 Number of 4-step king's tours on an n X n board summed over all starting positions.
%C A186863 From _J. Volkmar Schmidt_, Oct 25 2023 (Start)
%C A186863 Proof of formula for a(n) follows proof scheme from _David A. Corneth_ for A186864.
%C A186863 Distribution matrix of surrounding rectangles for 4-step walks is:
%C A186863   [0  0  0  2]
%C A186863   [0 24 80 28]
%C A186863   [0 80 80 20]
%C A186863   [2 28 20  4] (End)
%H A186863 J. Volkmar Schmidt, <a href="/A186863/b186863.txt">Table of n, a(n) for n = 1..50</a>
%H A186863 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3, -3, 1).
%F A186863 a(n) = 368*n^2 - 1308*n + 1108 = 4*(92*(n-1)*(n-3) + 41*n + 1) for n > 2.
%F A186863 G.f.: 4*x^2*(6 + 106*x + 87*x^2 - 15*x^3)/(1-x)^3. - _Colin Barker_, Jan 22 2012
%e A186863 Some solutions for 3 X 3:
%e A186863   0 3 0   0 2 0   0 0 1   1 0 0   0 0 0   0 4 3   4 0 0
%e A186863   0 2 4   0 3 1   0 0 2   4 2 0   0 1 4   0 2 1   1 3 0
%e A186863   0 1 0   0 0 4   4 3 0   3 0 0   0 3 2   0 0 0   2 0 0
%Y A186863 Row 4 of A186861.
%K A186863 nonn,easy
%O A186863 1,2
%A A186863 _R. H. Hardin_, Feb 27 2011
%E A186863 a(34)-a(39) from _J. Volkmar Schmidt_, Sep 03 2023