A275614 -id:A275614 - cp's OEIS frontend

A214070 Decimal expansion of the number whose continued fraction is 1, 2, 4, 8, 16, ...

1, 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

Offset: 1

Robert G. Wilson v, Aug 06 2012

Essentially the same as A096641. - R. J. Mathar, Aug 10 2012

			1.4459346405122026681195543406826176842704088452034385079032635605006619006916...

Cf. A000079, A096641, A275614.

Cf. A052119, A073824, A309930.

RealDigits[ FromContinuedFraction[{1, 2^Range@ 19}], 10, 111][[1]]

From Amiram Eldar, Feb 08 2022: (Start)
Equals A096641 + 1.
Equals 1/A275614. (End)

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

Original entry on oeis.org

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

Rick L. Shepherd, Jun 30 2004

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.

			0.445934640512202668119554340682617684270408845203438507903263560500661900...

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.

Cf. A155559, A214070, A275614.

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

\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 */

From Amiram Eldar, Feb 08 2022: (Start)
Equals A214070 - 1.
Equals 1/A275614 - 1. (End)

cp's OEIS Frontend

A214070 Decimal expansion of the number whose continued fraction is 1, 2, 4, 8, 16, ...

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