A169767 - OEIS

%I #13 Jul 22 2017 10:05:05

%S 0,0,0,0,0,0,0,0,0,0,16,0,0,0,124,0,0,0,1404,0,0,0,12824,0,0,0,126696,

%T 0,0,0,1222368,0,0,0,11930192,0,0,0,115974192,0,0,0,1128943296,0,0,0,

%U 10984783168,0,0,0,106897187552,0,0,0,1040241749856

%N Number of closed knight's tour diagrams of a 3 X n chessboard that have "Eulerian symmetry".

%C When the board is rotated 180 degrees, the diagram remains the same, and the second half of the tour is the same as the first half before rotation. (If the knight starts at one corner, he reaches the opposite corner after 3n/2 moves.)

%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="/A169767/b169767.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 A169767[n]=0 unless n mod 4 = 2.

%F Generating function: (2*(8*z^14 + 14*z^18 - 182*z^22 - 168*z^26 + 348*z^30 - 1000*z^34 + 13224*z^38 + 22904*z^42 - 105776*z^46 - 111616*z^50 + 292800*z^54 + 217536*z^58 - 294656*z^62 - 114432*z^66 - 22528*z^70 - 44032*z^74 + 180224*z^78 - 65536*z^82 + 32768*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 4,11

%A _N. J. A. Sloane_, May 10 2010, based on a communication from _Don Knuth_, Apr 28 2010