cp's OEIS Frontend

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

%I A096641 #24 Jan 14 2025 21:16:28
%S A096641 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,
%T A096641 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,
%U A096641 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
%N A096641 Decimal expansion of number with continued fraction expansion 0, 2, 4, 8, 16, ... (0 and positive powers of 2).
%C A096641 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.
%H A096641 James Mc Laughlin and Nancy J. Wyshinski, <a href="https://doi.org/10.1017/S0305004105008479">Ramanujan and the regular continued fraction expansion of real numbers</a>, Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 138. No. 3 (2005), pp. 367-381; <a href="https://arxiv.org/abs/math/0402461">arXiv preprint</a> arXiv:math/0402461 [math.NT], 2004; <a href="https://www.wcupa.edu/sciences-mathematics/mathematics/jMcLaughlin/documents/ramregcfsrealnos.pdf">alternative link</a>. See page 2.
%F A096641 From _Amiram Eldar_, Feb 08 2022: (Start)
%F A096641 Equals A214070 - 1.
%F A096641 Equals 1/A275614 - 1. (End)
%e A096641 0.445934640512202668119554340682617684270408845203438507903263560500661900...
%t A096641 RealDigits[FromContinuedFraction[{0, 2^Range@ 19}], 10, 111][[1]] (* _Robert G. Wilson v_, Jan 04 2013 *)
%o A096641 (PARI)
%o A096641 \p 400
%o A096641 dec_exp(v)= w=contfracpnqn(v); w[1,1]/w[2,1]+0.
%o A096641 dec_exp(vector(400,i,if(i==1,0,2^(i-1))))
%o A096641 /* The following uses Komatsu's expression for given a; a0=0, m=1 */
%o A096641 {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)))}
%o A096641 Komatsu(2) /* generates this sequence's constant */
%Y A096641 Cf. A155559, A214070, A275614.
%K A096641 cons,nonn
%O A096641 0,1
%A A096641 _Rick L. Shepherd_, Jun 30 2004