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A171155
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For two strings of length n, this is the number of pairwise alignments that do not have an insertion adjacent to a deletion.
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4
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1, 1, 3, 9, 27, 83, 259, 817, 2599, 8323, 26797, 86659, 281287, 915907, 2990383, 9786369, 32092959, 105435607, 346950321, 1143342603, 3772698725, 12463525229, 41218894577, 136451431723, 452116980643, 1499282161375, 4975631425581, 16524213199923, 54913514061867
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OFFSET
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0,3
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COMMENTS
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This is the number of walks from (0,0) to (n,n) where unit horizontal (+1,0), vertical (0,+1), and diagonal (+1,+1) steps are permitted but a horizontal step cannot be followed by a vertical step, nor vice-versa.
a(n) is also the number of walks from (0,0) to (n,n) with steps that increment one or two coordinates and having the property that no two consecutive steps are orthogonal. - Lee A. Newberg, Dec 04 2009
a(n) is also the number of standard sequence alignments of two 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 the first string is in the same column as some j-th position of the second string. - Lee A. Newberg, Dec 11 2009
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 0..1000
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FORMULA
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G.f.: sqrt((1-x)/(1-3*x-x^2-x^3)). - Mark van Hoeij, May 10 2013
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EXAMPLE
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For n = 3, the 9 alignments are:
ABC A-BC ABC- -ABC -ABC --ABC ABC- AB-C ABC--
DEF DEF- D-EF DEF- DE-F DEF-- -DEF -DEF --DEF
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MAPLE
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a:= proc(n) option remember; `if`(n<4, [1, 1, 3, 9][n+1],
((4*n-3)*a(n-1) -(2*n-5)*a(n-2) +a(n-3) -(n-3)*a(n-4))/n)
end:
seq (a(n), n=0..30); # Alois P. Heinz, Jan 22 2013
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PROG
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(PARI) x='x+O('x^66); Vec(sqrt((1-x)/(1-3*x-x^2-x^3))) \\ Joerg Arndt, May 11 2013
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CROSSREFS
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See A171158 for the number of such walks in three dimensions. - Lee A. Newberg, Dec 04 2009
See A171563 for the number of such walks in four dimensions. - Lee A. Newberg, Dec 11 2009
Sequence in context: A052917 A099786 A192909 * A131428 A099787 A176826
Adjacent sequences: A171152 A171153 A171154 * A171156 A171157 A171158
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KEYWORD
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nonn,walk,changed
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AUTHOR
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Lee A. Newberg, Dec 04 2009
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EXTENSIONS
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Extended beyond a(19) by Alois P. Heinz, Jan 22 2013
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STATUS
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approved
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