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A006466
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Continued fraction expansion of C = 2*sum( 1/2^(2^n), n=0 to infinity ).
(Formerly M0049)
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6
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1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| C arises when looking for a sequence b(n) such that : b(1)=0, b(n+1) is the smallest integer > b(n) such that the continued fraction for 1/2^b(1)+1/2^b(2)+...+1/2^b(n+1) contains only 1's or 2's. Because b(n)=2^n-1 and C = sum(k>=0,1/2^b(k)). - Benoit Cloitre (benoit7848c(AT)orange.fr), Nov 03 2002
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REFERENCES
| J. O. Shallit, Simple continued fractions for some irrational numbers. J. Number Theory 11 (1979), no. 2, 209-217.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| Harry J. Smith, Table of n, a(n) for n=0,...,20000
J. O. Shallit, Simple continued fractions for some irrational numbers. J. Number Theory 11 (1979), no. 2, 209-217.
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FORMULA
| Recurrence: a(5n)=a(5n+1)=a(2)=a(5n+3)=a(20n+14)=a(40n+9)=1, a(20n+4)=a(40n+29)=2, a(5n+2)=3-a(5n-1), a(20n+19)=a(10n+9). - Ralf Stephan (ralf(AT)ark.in-berlin.de), May 17 2005
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EXAMPLE
| 1.632843018043786287416159475... = 1 + 1/(1 + 1/(1 + 1/(1 + 1/(2 + ...)))) [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), May 09 2009]
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PROG
| (PARI) { allocatemem(932245000); default(realprecision, 10000); x=suminf(n=0, 1/2^(2^n)); x=contfrac(2*x); for (n=1, 20001, write("b006466.txt", n-1, " ", x[n])); } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), May 09 2009]
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CROSSREFS
| Cf. A076214 = Decimal expansion. [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), May 09 2009]
Sequence in context: A072911 A053150 A163379 * A086597 A031214 A056059
Adjacent sequences: A006463 A006464 A006465 * A006467 A006468 A006469
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KEYWORD
| nonn,cofr
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Better description and more terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jun 19 2001
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