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

Author

Rick L. Shepherd, Jun 30 2004

Keywords

• cons
• nonn

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.

Examples

			0.445934640512202668119554340682617684270408845203438507903263560500661900...
		

Links

• 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.

Crossrefs

Cf. A155559, A214070, A275614.

Programs

Mathematica

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

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 */

Formula

From Amiram Eldar, Feb 08 2022: (Start)

Equals A214070 - 1.

Equals 1/A275614 - 1. (End)