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

1,4

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

Formula

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

Asymptotic value: 0.02789*3.45059^n.

Crossrefs

Cf. A118067.

Sequence in context: A028701 A126270 A371665 * A192059 A191419 A054373

Adjacent sequences: A169693 A169694 A169695 * A169697 A169698 A169699

Keyword

nonn

Author

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

Status

approved