A169765 Number of closed knight's tour diagrams of a 3 X n chessboard that are symmetric with respect to left-right reflection about a vertical axis.

0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 24, 0, 0, 0, 276, 0, 0, 0, 2604, 0, 0, 0, 25736, 0, 0, 0, 248816, 0, 0, 0, 2424608, 0, 0, 0, 23581056, 0, 0, 0, 229513584, 0, 0, 0, 2233386048, 0, 0, 0, 21733496960, 0, 0, 0, 211495383968, 0, 0, 0, 2058092298080

Offset: 4

Author

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

Keywords

• nonn

Examples

			The first example, for n=10, was exhibited by Ernest Bergholt in British Chess Magazine 1918, page 74.
		

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 (terms 4..1002 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).

Crossrefs

Cf. A070030, A169696, A169764-A169777.

Formula

A169765(n)=0 unless n mod 4 = 2. And if n mod = 2, A169765(n) = A169765(n) + A169767(n).

Generating function: (2*(2*z^10 - 62*z^18 + 106*z^22 + 624*z^26 - 2560*z^30 - 2464*z^34 + 20640*z^38 + 11112*z^42 - 70304*z^46 - 75840*z^50 + 94976*z^54 + 206528*z^58 - 25216*z^62 - 60672*z^66 - 70656*z^70 - 168960*z^74 + 24576*z^78 + 81920*z^82 + 32768*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).

Extensions

More terms from Alois P. Heinz, Nov 26 2010