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Search: id:A005270
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| A005270 |
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Number of sequences s of length n with s[1]=1, s[2]=1, s[j-1]<s[j]<=s[j-2]+s[j-1] for j>=3.
(Formerly M1684)
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+0
3
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| 1, 1, 1, 2, 6, 27, 177, 1680, 23009, 455368, 13067353, 546378617, 33472296082, 3021920660821, 404374532614122, 80646410554881100, 24095492607316134304, 10837141045948365696938, 7369252748590790186483284, 11961418205662159081422825777494
(list; graph; listen)
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OFFSET
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2,4
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COMMENT
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The sequences of length n that are counted here are sub-Fibonacci sequences (A005269) with the property that its members, except for the initial two terms, strictly increase. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 15 2007
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Fishburn, Peter C. and Roberts, Fred S.; Elementary sequences, sub-Fibonacci sequences. Discrete Appl. Math. 44 (1993), no. 1-3, 261-281.
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FORMULA
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a(n) equals the number of nodes in generation n-2 of the sub-Fibonacci tree (A125051) for n>=2. - Paul D. Hanna (pauldhanna(AT)juno.com), Nov 19 2006
See the Maple program; g[k](x, y) is the number of sequences s[1], s[2], ..., s[k+2] such that s[1]=x, s[2]=y, s[j-1] <s[j] <= s[j-2]+s[j-1] for j>=3. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 15 2007
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EXAMPLE
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a(2)=6 because we have (1,1,2,3,4,5), (1,1,2,3,4,6), (1,1,2,3,4,7), (1,1,2,3,5,6), (1,1,2,3,5,7) and (1,1,2,3,5,8).
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MAPLE
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g[0]:=1:for k from 0 to 20 do g[k+1]:=expand(sum(subs({x=y, y=z}, g[k]), z=y+1..x+y)) od:seq(subs({x=1, y=1}, g[k]), k=0..20); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 15 2007
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CROSSREFS
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Cf. A125051, A125052.
Cf. A005269.
Sequence in context: A058712 A070076 A130455 this_sequence A080839 A118085 A011834
Adjacent sequences: A005267 A005268 A005269 this_sequence A005271 A005272 A005273
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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a(12) from Paul D. Hanna (pauldhanna(AT)juno.com), Nov 19 2006
Edited by Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 15 2007
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