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A096641
Decimal expansion of number with continued fraction expansion 0, 2, 4, 8, 16, ... (0 and positive powers of 2).
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
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
KEYWORD
cons,nonn
AUTHOR
Rick L. Shepherd, Jun 30 2004
STATUS
approved