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A171158
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The number of walks from (0,0,0) to (n,n,n) with steps that increment one to three coordinates and having the property that no two consecutive steps are orthogonal.
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3
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1, 1, 19, 235, 3181, 44725, 648439, 9614329, 145020445, 2217212539, 34269961873, 534449721793, 8397498847645, 132785160326593, 2111135363144743, 33723822603109987, 540949658114010583, 8708952402795685879, 140665766088396528829, 2278642960112808284773
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OFFSET
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0,3
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COMMENTS
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a(n) is also the number of standard sequence alignments of three strings of length n, counting only those alignments with the property that, for every pair of consecutive alignment columns, there is at least one sequence that contributes a non-gap to both columns. That is, a(n) counts only those standard alignments with a column order that can be unambiguously reconstructed from the knowledge of all pairings, where a pairing is, e.g., that some i-th position of some string x is in the same column as some j-th position of some string y. - Lee A. Newberg, Dec 11 2009
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LINKS
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FORMULA
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a(n) ~ c * d^n / n, where d = 17.073685937995..., c = 0.171212682922... . - Vaclav Kotesovec, Sep 10 2014
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EXAMPLE
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For n = 2, the 19 walks are:
000 -> 001 -> 012 -> 122 -> 222
000 -> 001 -> 102 -> 212 -> 222
000 -> 001 -> 112 -> 222
000 -> 010 -> 021 -> 122 -> 222
000 -> 010 -> 120 -> 221 -> 222
000 -> 010 -> 121 -> 222
000 -> 011 -> 112 -> 222
000 -> 011 -> 121 -> 222
000 -> 011 -> 122 -> 222
000 -> 100 -> 201 -> 212 -> 222
000 -> 100 -> 210 -> 221 -> 222
000 -> 100 -> 211 -> 222
000 -> 101 -> 112 -> 222
000 -> 101 -> 211 -> 222
000 -> 101 -> 212 -> 222
000 -> 110 -> 121 -> 222
000 -> 110 -> 211 -> 222
000 -> 110 -> 221 -> 222
000 -> 111 -> 222
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CROSSREFS
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See A171155 for the number of such walks in two dimensions.
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KEYWORD
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nonn,walk
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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