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A096641 Decimal expansion of number with continued fraction expansion 0, 2, 4, 8, 16, ... (0 and positive powers of 2). 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, 4 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

According to the McLaughlin-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.

J. McLaughlin and N. J. Wyshinski, Ramanujan and the Regular Continued Fraction Expansion of Real Numbers, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 138, Issue 3 May 2005, pp. 367-381, see page 2. [See also https://arxiv.org/abs/math/0402461]

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. A214070.

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

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Last modified March 29 21:48 EDT 2017. Contains 284288 sequences.