A076214 Decimal expansion of C = Sum_{k>=0} 1/2^(2^k-1).

1, 6, 3, 2, 8, 4, 3, 0, 1, 8, 0, 4, 3, 7, 8, 6, 2, 8, 7, 4, 1, 6, 1, 5, 9, 4, 7, 5, 0, 6, 1, 0, 5, 0, 4, 4, 3, 4, 0, 6, 6, 2, 2, 7, 5, 1, 8, 4, 1, 1, 0, 5, 6, 0, 8, 6, 8, 2, 4, 2, 1, 8, 0, 7, 6, 8, 6, 1, 1, 1, 2, 2, 8, 3, 8, 9, 1, 1, 0, 6, 0, 0, 1, 2, 0, 9, 7, 0, 6, 2, 6, 4, 9, 6, 7, 9, 4, 5, 3, 1, 2, 3, 5, 1, 1

Offset: 1

Author

Benoit Cloitre, Nov 03 2002

Keywords

• cons
• nonn

Comments

This constant has a nice continued fraction expansion (i.e. only 1 and 2 occur). 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).

Examples

			1.632843018043786287416159475061050443406622751841105608682421807686111...
		

Links

• Harry J. Smith, Table of n, a(n) for n = 1..20000
• Boris Adamczewski, The Many Faces of the Kempner Number, Journal of Integer Sequences, Vol. 16 (2013), Article 13.2.15.
• Michael Ian Shamos, Property Enumerators and a Partial Sum Theorem, 2011; alternative link.

Crossrefs

Cf. A006466 (continued fraction), A007404, A078585.

Programs

Mathematica

Take[ RealDigits[ 2*NSum[1/2^2^k, {k, 0, Infinity}, WorkingPrecision -> 120]][[1]], 105] (* Jean-François Alcover, Nov 15 2011 *)

PARI

default(realprecision, 20080); x=suminf(k=0, 1/2^(2^k)); x*=2; for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b076214.txt", n, " ", d)); \\ Harry J. Smith, May 09 2009

Formula

Equals 2 * Sum_{k>=0} 1/2^(2^k) = 2 * A007404. - Harry J. Smith, May 09 2009

From Amiram Eldar, Mar 12 2024: (Start)

Equals 1 + 2 * A078585.

Equals 1 + Sum_{k>=1} floor(log_2(k))/2^k (Shamos, 2011, p. 8). (End)