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A169765 Number of closed knight's tour diagrams of a 3 X n chessboard that are symmetric with respect to left-right reflection about a vertical axis.

%I A169765 #23 Jul 22 2017 10:04:54
%S A169765 0,0,0,0,0,0,4,0,0,0,24,0,0,0,276,0,0,0,2604,0,0,0,25736,0,0,0,248816,
%T A169765 0,0,0,2424608,0,0,0,23581056,0,0,0,229513584,0,0,0,2233386048,0,0,0,
%U A169765 21733496960,0,0,0,211495383968,0,0,0,2058092298080
%N A169765 Number of closed knight's tour diagrams of a 3 X n chessboard that are symmetric with respect to left-right reflection about a vertical axis.
%D A169765 D. E. Knuth, Long and skinny knight's tours, in Selected Papers on Fun and Games, to appear, 2010.
%H A169765 Seiichi Manyama, <a href="/A169765/b169765.txt">Table of n, a(n) for n = 4..4057</a> (terms 4..1002 from Alois P. Heinz)
%H A169765 George Jelliss, <a href="http://www.mayhematics.com/t/oa.htm">Open knight's tours of three-rank boards</a>, Knight's Tour Notes, note 3a (21 October 2000).
%H A169765 George Jelliss, <a href="http://www.mayhematics.com/t/ob.htm">Closed knight's tours of three-rank boards</a>, Knight's Tour Notes, note 3b (21 October 2000).
%F A169765 A169765(n)=0 unless n mod 4 = 2. And if n mod = 2, A169765(n) = A169765(n) + A169767(n).
%F A169765 Generating function: (2*(2*z^10 - 62*z^18 + 106*z^22 + 624*z^26 - 2560*z^30 - 2464*z^34 + 20640*z^38 + 11112*z^42 - 70304*z^46 - 75840*z^50 + 94976*z^54 + 206528*z^58 - 25216*z^62 - 60672*z^66 - 70656*z^70 - 168960*z^74 + 24576*z^78 + 81920*z^82 + 32768*z^86))/
%F A169765 (1 - 6*z^4 - 64*z^8 + 200*z^12 + 1000*z^16 - 3016*z^20 - 3488*z^24 + 24256*z^28 - 23776*z^32 - 104168*z^36 + 203408*z^40 + 184704*z^44 - 443392*z^48 - 14336*z^52 + 151296*z^56 - 145920*z^60 + 263424*z^64 - 317440*z^68 - 36864*z^72 + 966656*z^76 - 573440*z^80 - 131072*z^84).
%e A169765 The first example, for n=10, was exhibited by Ernest Bergholt in British Chess Magazine 1918, page 74.
%Y A169765 Cf. A070030, A169696, A169764-A169777.
%K A169765 nonn
%O A169765 4,7
%A A169765 _N. J. A. Sloane_, May 10 2010, based on a communication from _Don Knuth_, Apr 28 2010
%E A169765 More terms from _Alois P. Heinz_, Nov 26 2010