A006466 Continued fraction expansion of C = 2*Sum_{n>=0} 1/2^(2^n). (Formerly M0049)

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

Harry J. Smith, Table of n, a(n) for n = 0..20000

Boris Adamczewski, The Many Faces of the Kempner Number, Journal of Integer Sequences, Vol. 16 (2013), #13.2.15.

J. Shallit, Letter to N. J. A. Sloane with attachment, Aug. 1979

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

Adjacent sequences: A006463 A006464 A006465 * A006467 A006468 A006469

Keyword

nonn,cofr

Author

N. J. A. Sloane

Extensions

Better description and more terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jun 19 2001

Status

approved