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A233590 Decimal expansion of the continued fraction c(1) +c(1)/(c(2) +c(2)/(c(3) +c(3)/(c(4) +c(4)/....))), where c(i)=2^(i-1).

%I A233590 #39 Oct 26 2024 22:59:11
%S A233590 1,4,0,8,6,1,5,9,7,9,7,3,5,0,0,5,2,0,5,1,3,2,3,6,2,5,9,0,2,5,5,7,9,5,
%T A233590 2,0,9,4,8,4,5,6,3,3,7,3,6,8,6,8,8,8,3,5,3,7,0,3,9,2,7,0,2,2,3,7,9,7,
%U A233590 5,9,9,8
%N A233590 Decimal expansion of the continued fraction c(1) +c(1)/(c(2) +c(2)/(c(3) +c(3)/(c(4) +c(4)/....))), where c(i)=2^(i-1).
%C A233590 For more details about this type of continued fraction, see A233588.
%C A233590 This one corresponds to the powers of two sequence.
%C A233590 Corresponds to the regular continued fraction 1,2,2,4,4,8,8,16,16,... = A060546. - _Jeffrey Shallit_, Jun 14 2016
%H A233590 Stanislav Sykora, <a href="/A233590/b233590.txt">Table of n, a(n) for n = 1..20000</a> [a(1522) onward corrected by _Kevin Ryde_, Oct 26 2024]
%H A233590 Stanislav Sykora, <a href="http://dx.doi.org/10.3247/sl4math13.001">Blazys' Expansions and Continued Fractions</a>, Stans Library, Vol.IV, 2013, DOI 10.3247/sl4math13.001
%H A233590 Stanislav Sykora, <a href="http://oeis.org/wiki/File:BlazysExpansions.txt">PARI/GP scripts for Blazys expansions and fractions</a>, OEIS Wiki
%F A233590 Equals 1+1/(2+2/(4+4/(8+8/(16+16/(32+...))))).
%F A233590 Equals Product_{k>=0} ((1 - 2^(5*k + 2))*(1 - 2^(5*k + 3)))/((1 - 2^(5*k + 1))*(1 - 2^(5*k + 4))). - _Antonio Graciá Llorente_, Mar 20 2024
%e A233590 1.408615979735005205132362590255795209484563373686888353703927022...
%t A233590 RealDigits[ Fold[(#2 + #2/#1) &, 1, Reverse@ (2^Range[0, 27])], 10, 111][[1]] (* _Robert G. Wilson v_, May 22 2014 *)
%o A233590 (PARI) See the link
%Y A233590 Cf. A000079 (2^n), A096658, A060546.
%Y A233590 Cf. Blazys's continued fractions: A233588, A233589, A233591 and Blazys' expansions: A233582, A233583, A233584, A233585, A233586, A233587
%K A233590 nonn,cons
%O A233590 1,2
%A A233590 _Stanislav Sykora_, Jan 06 2014