A169766 - OEIS

%I #13 Jul 22 2017 12:56:13

%S 0,0,0,0,0,0,4,0,0,0,8,0,48,0,152,0,352,0,1200,0,3752,0,12912,0,34768,

%T 0,122120,0,346128,0,1202240,0,3337424,0,11650864,0,32634960,0,

%U 113539392,0,316870592,0,1104442752,0,3086894528,0,10748713792,0,30023935744,0

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

%C When the board is rotated 180 degrees, the diagram remains the same, but the tour reverses direction.

%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="/A169766/b169766.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) = 0 unless n mod 2 = 0.

%F Generating function: (2*(2*z^10 - 8*z^14 + 24*z^16 - 76*z^18 + 32*z^20 + 288*z^22 - 716*z^24 + 792*z^26 - 336*z^28 - 2908*z^30 + 7896*z^32 - 1464*z^34 - 3432*z^36 + 7416*z^38 - 32616*z^40 - 11792*z^42 + 39888*z^44 + 35472*z^46 + 47968*z^48 + 35776*z^50 - 143424*z^52 - 197824*z^54 - 15552*z^56 - 11008*z^58 + 181376*z^60 + 269440*z^62 + 78080*z^64 + 53760*z^66 + 44288*z^68 - 48128*z^70 - 112640*z^72 - 124928*z^74 - 227328*z^76 - 155648*z^78 + 98304*z^80 + 147456*z^82 + 32768*z^84))/

%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

%E More terms extracted from the g.f. by _R. J. Mathar_, Oct 09 2010