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Search: id:A096121
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| A096121 |
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Number of full spectrum rook's walks on a (2 X n) board. |
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+0
1
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| 2, 8, 60, 816, 17520, 550080, 23839200, 1365799680, 100053999360, 9127781913600, 1015061950425600, 135193044668774400, 21248464632595200000, 3891825697262043340800, 821745573997874093568000
(list; graph; listen)
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OFFSET
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1,1
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REFERENCES
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A rook must land on each square exactly once, but may start and end anywhere and may intersect its own path. Inspired by Leroy Quet in a Jul 05 2004 posting to the Seqfan mailing list.
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LINKS
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Leroy Quet, Home Page (listed in lieu of email address)
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FORMULA
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a(n+1)=n(n+1)(a(n) + a(n-1)) for n > 1
Further refinement gives: a(n + 1) = 2(n + 1)! sum_{k = 0}^{floor(n / 2)} { perm(n - k, k ) * comb(n - k, k) + perm(n - k, k + 1) * comb(n - 1 - k, i) } (where perm(a, b) == a!/b!; comb(a, b) == a!/b!/(a-b)!).
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EXAMPLE
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Tagging the squares on a (3 X 2) board with A,B,C/D,E,F, the 10 tours starting at A are: ABCFDE, ABCFED, ABEDFC, ACBEDF, ACBEFD, ACFDEB, ADEBCF, ADEFCB, ADFCBE, ADFEBC. There are a similar 10 tours starting at each of the other 5 squares, so a(3) = 6 * 10 = 60.
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CROSSREFS
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Cf. A096970 and references to "rook tours" or "rook walks".
Sequence in context: A001188 A113145 A036794 this_sequence A143217 A139017 A085657
Adjacent sequences: A096118 A096119 A096120 this_sequence A096122 A096123 A096124
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KEYWORD
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nonn,easy,walk
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AUTHOR
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hv(AT)crypt.org (Hugo van der Sanden), Jul 22 2004
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