A169769 Number of geometrically distinct closed knight's tours of a 3 X n chessboard.
0, 0, 0, 0, 0, 0, 6, 0, 44, 0, 396, 0, 3868, 0, 37070, 0, 362192, 0, 3516314, 0, 34237842, 0, 333077332, 0, 3241403380, 0, 31542464952, 0, 306944118820, 0, 2986962829456, 0, 29066627247828, 0, 282854730020224, 0, 2752516325518516, 0
Offset: 4
Keywords
Examples
The six solutions for n=10 were first published by Kraitchik in 1927.
References
- D. E. Knuth, Long and skinny knight's tours, in Selected Papers on Fun and Games, to appear, 2010.
Links
- Seiichi Manyama, Table of n, a(n) for n = 4..2031 (terms 4..1000 from Alois P. Heinz)
- George Jelliss, Open knight's tours of three-rank boards, Knight's Tour Notes, note 3a (21 October 2000).
- George Jelliss, Closed knight's tours of three-rank boards, Knight's Tour Notes, note 3b (21 October 2000).
Formula
a(n) = 0 unless n mod 2 = 0.
Generating function: 2*z^10*((-2*(1 + 5*z^2 - 34*z^4 - 116*z^6 + 505*z^8 + 616*z^10 - 3179*z^12 - 4*z^14 + 9536*z^16 - 8176*z^18 - 13392*z^20 + 15360*z^22 + 13888*z^24 + 2784*z^26 - 3328*z^28 - 22016*z^30 + 5120*z^32 + 2048*z^34))/
(-1 + 6*z^2 + 64*z^4 - 200*z^6 - 1000*z^8 + 3016*z^10 + 3488*z^12 - 24256*z^14 + 23776*z^16 + 104168*z^18 - 203408*z^20 - 184704*z^22 + 443392*z^24 + 14336*z^26 - 151296*z^28 + 145920*z^30 - 263424*z^32 + 317440*z^34 + 36864*z^36 - 966656*z^38 + 573440*z^40 + 131072*z^42) -
(1 + 6*z^6 - 31*z^8 + 8*z^10 + 53*z^12 - 179*z^14 + 312*z^16 - 84*z^18 - 1280*z^20 + 1974*z^22 - 1232*z^24 - 858*z^26 + 10320*z^28 - 8154*z^30 + 5556*z^32 + 9972*z^34 - 35152*z^36 + 11992*z^38 - 37920*z^40 - 35856*z^42 + 47488*z^44 - 3888*z^46 + 103264*z^48 + 45344*z^50 - 12608*z^52 + 19520*z^54 - 30336*z^56 + 11072*z^58 - 35328*z^60 - 28160*z^62 - 84480*z^64 - 56832*z^66 + 12288*z^68 + 24576*z^70 + 40960*z^72 + 8192*z^74 + 16384*z^76)/
(-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)).
Extensions
More terms from R. J. Mathar, Oct 09 2010