A169765 - OEIS

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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

(list; graph; refs; listen; history; text; internal format)

OFFSET

4,7

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).

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).

EXAMPLE

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

CROSSREFS

Cf. A070030, A169696, A169764-A169777.

Sequence in context: A265831 A348732 A264769 * A169768 A057386 A099306

Adjacent sequences: A169762 A169763 A169764 * A169766 A169767 A169768

KEYWORD

nonn

AUTHOR

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

EXTENSIONS

More terms from Alois P. Heinz, Nov 26 2010

STATUS

approved