A096641 Decimal expansion of number with continued fraction expansion 0, 2, 4, 8, 16, ... (0 and positive powers of 2).

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

Offset

0,1

Comments

According to the Mc Laughlin-Wyshinski paper, Tasoev proposed continued fractions of the form [a0;a,...,a,a^2,...,a^2,a^3,...,a^3,...], where a0 >= 0, a >= 2 and m >= 1 are integers and each power of a occurs m times. This sequence is for the minimal values a0 = 0, a = 2 and m = 1. Komatsu "derived a closed form for the general case (m >= 1, arbitrary)" and the expression given in (1.2) (where a0=0 and m=1) of the linked paper and which is used in the second PARI/GP program below.

Links

Table of n, a(n) for n=0..104.

James Mc Laughlin and Nancy J. Wyshinski, Ramanujan and the regular continued fraction expansion of real numbers, Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 138. No. 3 (2005), pp. 367-381; arXiv preprint arXiv:math/0402461 [math.NT], 2004; alternative link. See page 2.

Formula

From Amiram Eldar, Feb 08 2022: (Start)

Equals A214070 - 1.

Equals 1/A275614 - 1. (End)

Example

0.445934640512202668119554340682617684270408845203438507903263560500661900...

Mathematica

RealDigits[FromContinuedFraction[{0, 2^Range@ 19}], 10, 111][[1]] (* Robert G. Wilson v, Jan 04 2013 *)

Prog

(PARI)
\p 400
dec_exp(v)= w=contfracpnqn(v); w[1, 1]/w[2, 1]+0.
dec_exp(vector(400, i, if(i==1, 0, 2^(i-1)))
/* The following uses Komatsu's expression for given a; a0=0, m=1 */
{Komatsu(a)=suminf(s=0, a^(-(s+1)^2)*prod(i=1, s, (a^(2*i)-1)^(-1))) /suminf(s=0, a^(-s^2)*prod(i=1, s, (a^(2*i)-1)^(-1)))}
Komatsu(2) /* generates this sequence's constant */

Crossrefs

Cf. A155559, A214070, A275614.

Sequence in context: A072231 A281375 A214070 * A155693 A160705 A107851

Adjacent sequences: A096638 A096639 A096640 * A096642 A096643 A096644

Keyword

cons,nonn

Author

Rick L. Shepherd, Jun 30 2004

Status

approved