

A169767


Number of closed knight's tour diagrams of a 3 X n chessboard that have "Eulerian symmetry".


3



0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16, 0, 0, 0, 124, 0, 0, 0, 1404, 0, 0, 0, 12824, 0, 0, 0, 126696, 0, 0, 0, 1222368, 0, 0, 0, 11930192, 0, 0, 0, 115974192, 0, 0, 0, 1128943296, 0, 0, 0, 10984783168, 0, 0, 0, 106897187552, 0, 0, 0, 1040241749856
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OFFSET

4,11


COMMENTS

When the board is rotated 180 degrees, the diagram remains the same, and the second half of the tour is the same as the first half before rotation. (If the knight starts at one corner, he reaches the opposite corner after 3n/2 moves.)


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 threerank boards, Knight's Tour Notes, note 3a (21 October 2000).
George Jelliss, Closed knight's tours of threerank boards, Knight's Tour Notes, note 3b (21 October 2000).


FORMULA

A169767[n]=0 unless n mod 4 = 2.
Generating function: (2*(8*z^14 + 14*z^18  182*z^22  168*z^26 + 348*z^30  1000*z^34 + 13224*z^38 + 22904*z^42  105776*z^46  111616*z^50 + 292800*z^54 + 217536*z^58  294656*z^62  114432*z^66  22528*z^70  44032*z^74 + 180224*z^78  65536*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).


CROSSREFS

Cf. A070030, A169696, A169764A169777.
Sequence in context: A181029 A072838 A023919 * A225611 A173293 A008433
Adjacent sequences: A169764 A169765 A169766 * A169768 A169769 A169770


KEYWORD

nonn


AUTHOR

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


STATUS

approved



