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%I #46 Jan 21 2024 15:29:26
%S 1,5,4,1,4,9,4,0,8,2,5,3,6,7,9,8,2,8,4,1,3,1,1,0,3,4,4,4,4,7,2,5,1,4,
%T 6,3,8,3,4,0,4,5,9,2,3,6,8,4,1,8,8,2,1,0,9,4,7,4,1,3,6,9,5,6,6,3,7,5,
%U 4,2,6,3,9,1,4,3,3,1,4,8,0,7,0,7,1,8,2,5,7,2,4,0,8,5,0,0,7,7,4,2,2,4
%N Decimal expansion of 1/(sqrt(e) - 1).
%C Has continued fraction 1+2/(3+4/(5+6/7+...)).
%C Simple continued fraction is 1, 1, 1, 5, 1, 1, 9, 1, 1, 13, 1, 1, 17, 1, {1, 4k+1, 1}, ..., . - _Robert G. Wilson v_, Jul 01 2007
%H G. C. Greubel, <a href="/A113011/b113011.txt">Table of n, a(n) for n = 1..20000</a>
%H Leonhard Euler, <a href="https://arxiv.org/abs/math/0508227">On the formation of continued fractions</a>, arXiv:math/0508227 [math.HO], 2005, see p. 14.
%H Michel Waldschmidt, <a href="http://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/ContinuedFractionsOujda2015.pdf">Continued fractions</a>, Ecole de recherche CIMPA-Oujda, Théorie des Nombres et ses Applications, 18 - 29 mai 2015: Oujda (Maroc).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ContinuedFraction.html">Continued Fraction</a>.
%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F Equals Integral_{x = 0..oo} floor(2*x)*exp(-x) dx. - _Peter Bala_, Oct 09 2013
%F Equals 3/2 + Sum_{k>=0} tanh(1/2^(k+3))/2^(k+2). - _Antonio Graciá Llorente_, Jan 21 2024
%e 1.54149408253679828413110344447251463834045923684188210947413695663...
%t First@ RealDigits[ 1 / (Exp[1/2] - 1), 10, 111] (* _Robert G. Wilson v_, Jul 01 2007 *)
%t f[n_] := Fold[ Last@ #2 + First@ #2/#1 &, 2n - 1, Partition[ Reverse@ Range[ 2n - 2], 2]]; RealDigits[ f[61], 10, 105][[1]] (* _Robert G. Wilson v_, Jul 07 2012 *)
%o (PARI) 1/(sqrt(exp(1)) - 1) \\ _G. C. Greubel_, Apr 09 2018
%o (Magma) 1/(Sqrt(Exp(1)) - 1); // _G. C. Greubel_, Apr 09 2018
%Y Cf. A113012, A113013.
%K nonn,cons
%O 1,2
%A _Eric W. Weisstein_, following a suggestion of Grover W. Hughes, Oct 09 2005
%E Simpler definition from _T. D. Noe_, Oct 09 2005
%E Euler reference from _David L. Harden_, Oct 09 2005
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