A214070 -id:A214070 - cp's OEIS frontend

A073824 Decimal expansion of number with continued fraction expansion 0, 1, 4, 9, ... (the squares).

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

Offset: 0

Rick L. Shepherd, Aug 12 2002

			0.80431856111715795076768044139...

Kerry Mitchell, Table of n, a(n) for n = 0..1494

Cf. A000290 (squares), A052119, A309930, A214070.

The numerators and denominators of convergents to this constant are given by A036245 and A036246 respectively.

RealDigits[FromContinuedFraction[Range[0,100]^2],10,120][[1]] (* Harvey P. Dale, May 07 2018 *)

dec_exp(v)= w=contfracpnqn(v); w[1,1]/w[2,1]+0.
dec_exp(vector(2000,i,(i-1)^2))

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)

A275614 Decimal expansion of number with continued fraction expansion 0, 1, 2, 4, 8, 16, 32, ... (powers of 2).

Original entry on oeis.org

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

Offset: 0

José Eduardo Gaboardi de Carvalho, Aug 03 2016

			0.69159419242197808428289286692643...

Cf. A000079, A096641, A214070.

terms=sqrtint(bitprecision(1.))+2; \\ terms needed at current precision
t=contfracpnqn(vector(terms,i,.5<Charles R Greathouse IV, Aug 04 2016

From Amiram Eldar, Feb 08 2022: (Start)
Equals 1/A214070.
Equals 1/(1 + A096641). (End)

Definition corrected by Jianing Song, Mar 29 2025

cp's OEIS Frontend

A073824 Decimal expansion of number with continued fraction expansion 0, 1, 4, 9, ... (the squares).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A275614 Decimal expansion of number with continued fraction expansion 0, 1, 2, 4, 8, 16, 32, ... (powers of 2).

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Formula

Extensions