A169696 Number of undirected Knight's tours on a 3 X n board.

0, 0, 0, 8, 0, 0, 52, 396, 560, 3048, 10672, 57248, 128864, 646272, 1838784, 8636880, 23400992, 105865688, 305753680, 1322849752, 3862974304, 16225820000, 48744080192, 198673312880, 607041217056, 2417584484232, 7519864632928, 29320809649000, 92507134938336

Offset: 1

Author

N. J. A. Sloane, Apr 14 2010, based on a communication from Don Knuth

Keywords

• nonn

Comments

I think the (old) name "Number of open Knight's tours on a 3 X n board" is somewhat incorrect, because included are those tours in which the start/end cells are knight-neighbors. Such tours are potentially closed, although actually closing them would deprive them of specific start/end cells. "Number of undirected Knight's tours on a 3 X n board" would be a better name. For example the 3x10 has 3048 undirected tours, which would be 6096 directed tours, in accord with Colin Rose results (http://www.tri.org.au/knightframe.html, Solutions:3xm). Note that the 3x10 also has 16 closed tours (A169764 Number of closed Knight's tours on a 3 X n board), and each of those closed tour appears 30 times among the 3048 undirected tours, and 60 times among the 6096 directed tours. - Pierre Charland, Feb 15 2011

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 = 1..1861
• 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).
• D. E. Knuth, Comments, generating function, first 100 terms

Crossrefs

Cf. A118067.

Formula

a(n) = A169770(n) + A169771(n) + A169772(n).

Asymptotic value: 0.02789*3.45059^n.