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A169767
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Number of closed knight's tour diagrams of a 3 X n chessboard that have "Eulerian symmetry".
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3
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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
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4,11
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COMMENTS
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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.)
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REFERENCES
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D. E. Knuth, Long and skinny knight's tours, in Selected Papers on Fun and Games, to appear, 2010.
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LINKS
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FORMULA
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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).
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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