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Search: id:A001030
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| A001030 |
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Fixed under 1 -> 21, 2 -> 211.
(Formerly M0068 N0021)
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+0
6
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| 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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If treated as the terms of a continued fraction, it converges to approximately
2.57737020881617828717350576260723346479894963737498275232531856357441\
7024804797827856956758619431996.
The g.f. (-2-z**2-z**4-z-2*z**3-z**7+z**8)/(z-1)/(z**4+z**3+z**2+z+1) conjectured by S. Plouffe in his 1992 dissertation is wrong since it does not match all the terms. - M. F. Hasler, May 12 2008
There are a(n) 1's between successive 2's [From Eric Angelini (eric.angelini(AT)skynet.be), Aug 19 2008]
Same sequence where 1's and 2's are exchanged: A001468 [From Eric Angelini (eric.angelini(AT)skynet.be), Aug 19 2008]
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REFERENCES
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Midhat J. Gazale, Number: From Ahmes to Cantor, Section on 'Cleavages' in Chapter 6, Princeton University Press, Princeton, NJ 2000, pp. 203-211.
A. Nagel, A self-defining infinite sequence, with an application to Markoff chains and probability, Math. Mag., 36 (1963), 179-183.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..8119
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
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FORMULA
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a(n)= -1 + floor(n*(1+sqrt(2))+1/sqrt(2))-floor((n-1)*(1+sqrt(2))+1/sqrt(2)). - Benoit Cloitre, Jun 26 2004. [I don't know if this is a theorem or a conjecture. - N. J. A. Sloane (njas(AT)research.att.com), May 14 2008.]
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MATHEMATICA
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('n' is the number of substitution steps to perform.) Nest[Flatten[ # /. {1 -> {2, 1}, 2 -> {2, 1, 1}}] &, {1}, n]
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PROGRAM
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A001030 := proc(n) begin [ 2 ]; while nops(%)<n do subs(%, [ 1=(2, 1), 2=(2, 1, 1) ]) end_while; %[ n ] end_proc:
Contribution from K. Spage (kevspage2001(AT)yahoo.co.uk), Oct 08 2009: (Start)
(PARI) /*Fast string concatenation method giving e.g. 5740 terms in 8 iterations*/
a="2"; b="2, 1, 1, 2"; print1(b); for(x=1, 8, c=concat([", 1, ", a, ", 1, ", b]); print1(c); a=b; b=concat(b, c)) (End)
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CROSSREFS
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Length of the sequence after 'n' substitution steps is given by the terms of A000129.
Equals A004641(n) + 1.
Sequence in context: A103921 A115623 A134265 this_sequence A071709 A131406 A029440
Adjacent sequences: A001027 A001028 A001029 this_sequence A001031 A001032 A001033
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KEYWORD
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nonn,nice,easy
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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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More terms and additional comments from Peter Bertok (peter(AT)bertok.com), Nov 27 2001
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