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A006466
Continued fraction expansion of C = 2*Sum_{n>=0} 1/2^(2^n).
(Formerly M0049)
8
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
OFFSET
0,5
COMMENTS
C arises when looking for a sequence b(n) such that b(1)=0 and 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. It arises because b(n) = 2^n - 1 and C = Sum_{k>=0} 1/2^b(k). - Benoit Cloitre, Nov 03 2002
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Boris Adamczewski, The Many Faces of the Kempner Number, Journal of Integer Sequences, Vol. 16 (2013), #13.2.15.
Jeffrey Shallit, Simple continued fractions for some irrational numbers. J. Number Theory 11 (1979), no. 2, 209-217 [DOI]
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, May 17 2005
EXAMPLE
1.632843018043786287416159475... = 1 + 1/(1 + 1/(1 + 1/(1 + 1/(2 + ...)))). - Harry J. Smith, May 09 2009
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])); } \\ Harry J. Smith, May 09 2009
CROSSREFS
Cf. A076214 = Decimal expansion. - Harry J. Smith, May 09 2009
Sequence in context: A053150 A295657 A163379 * A316439 A335078 A086597
KEYWORD
nonn,cofr
EXTENSIONS
Better description and more terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jun 19 2001
STATUS
approved