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%I #14 Jul 22 2017 12:56:28
%S 0,0,0,0,0,0,4,0,0,0,24,0,24,0,276,0,176,0,2604,0,1876,0,25736,0,
%T 17384,0,248816,0,173064,0,2424608,0,1668712,0,23581056,0,16317480,0,
%U 229513584,0,158435296,0,2233386048,0,1543447264,0,21733496960
%N Number of geometrically distinct closed knight's tours of a 3 X n chessboard that have twofold symmetry.
%D D. E. Knuth, Long and skinny knight's tours, in Selected Papers on Fun and Games, to appear, 2010.
%H Seiichi Manyama, <a href="/A169768/b169768.txt">Table of n, a(n) for n = 4..4057</a>
%H 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 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 a(n) = (A169765(n)+A169766(n)+A169767(n))/2.
%F a(n) = 0 unless n mod 2 = 0.
%F Generating function: (4*z^10 + 24*z^16 - 124*z^18 + 32*z^20 + 212*z^22 - 716*z^24 + 1248*z^26 - 336*z^28 - 5120*z^30 + 7896*z^32 - 4928*z^34 - 3432*z^36 + 41280*z^38 - 32616*z^40 + 22224*z^42 + 39888*z^44 - 140608*z^46 + 47968*z^48 - 151680*z^50 - 143424*z^52 + 189952*z^54 - 15552*z^56 + 413056*z^58 + 181376*z^60 - 50432*z^62 + 78080*z^64 - 121344*z^66 + 44288*z^68 - 141312*z^70 - 112640*z^72 - 337920*z^74 - 227328*z^76 + 49152*z^78 + 98304*z^80 + 163840*z^82 + 32768*z^84 + 65536*z^86)/
%F (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).
%Y Cf. A070030, A169696, A169764-A169777.
%K nonn
%O 4,7
%A _N. J. A. Sloane_, May 10 2010, based on a communication from _Don Knuth_, Apr 28 2010
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