A169768 Number of geometrically distinct closed knight's tours of a 3 X n chessboard that have twofold symmetry.

0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 24, 0, 24, 0, 276, 0, 176, 0, 2604, 0, 1876, 0, 25736, 0, 17384, 0, 248816, 0, 173064, 0, 2424608, 0, 1668712, 0, 23581056, 0, 16317480, 0, 229513584, 0, 158435296, 0, 2233386048, 0, 1543447264, 0, 21733496960

Offset: 4

Author

N. J. A. Sloane, May 10 2010, based on a communication from Don Knuth, Apr 28 2010

Keywords

• nonn

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..4057
• 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).

Crossrefs

Cf. A070030, A169696, A169764-A169777.

Formula

a(n) = (A169765(n)+A169766(n)+A169767(n))/2.

a(n) = 0 unless n mod 2 = 0.

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)/

(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).