This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A113011 #59 Feb 16 2025 08:32:59
%S A113011 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 A113011 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 A113011 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 A113011 Decimal expansion of 1/(sqrt(e) - 1).
%C A113011 Has continued fraction 1+2/(3+4/(5+6/7+...)).
%C A113011 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 A113011 G. C. Greubel, <a href="/A113011/b113011.txt">Table of n, a(n) for n = 1..20000</a>
%H A113011 Itay Beit-Halachmi and Ido Kaminer, <a href="https://arxiv.org/abs/2412.12361">The Ramanujan Library -- Automated Discovery on the Hypergraph of Integer Relations</a>, arXiv:2412.12361 [cs.AI], 2024. See p. 6.
%H A113011 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 A113011 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 A113011 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ContinuedFraction.html">Continued Fraction</a>.
%H A113011 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%F A113011 Equals Integral_{x = 0..oo} floor(2*x)*exp(-x) dx. - _Peter Bala_, Oct 09 2013
%F A113011 Equals 3/2 + Sum_{k>=0} tanh(1/2^(k+3))/2^(k+2). - _Antonio Graciá Llorente_, Jan 21 2024
%F A113011 Conjecture: 1/(sqrt(e) - 1) = 1 + K_{n>=1} 2*n/(4*n^2-1)/1, where K is the Gauss notation for an infinite continued fraction. In the expanded form, 1/(sqrt(e) - 1) = 1 + 2/3/(1 + 4/15/(1 + 6/35/(1 + ...))) (see Beit-Halachmi and Kaminer). - _Stefano Spezia_, Dec 27 2024
%F A113011 Equals 1/(A019774 - 1). - _Hugo Pfoertner_, Dec 27 2024
%e A113011 1.54149408253679828413110344447251463834045923684188210947413695663...
%t A113011 First@ RealDigits[ 1 / (Exp[1/2] - 1), 10, 111] (* _Robert G. Wilson v_, Jul 01 2007 *)
%t A113011 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 *)
%t A113011 Rest[realDigitsRecip[Sqrt[E]-1]] (* The realDigitsRecip program is at A021200 *) (* _Harvey P. Dale_, Nov 04 2024 *)
%o A113011 (PARI) 1/(sqrt(exp(1)) - 1) \\ _G. C. Greubel_, Apr 09 2018
%o A113011 (Magma) 1/(Sqrt(Exp(1)) - 1); // _G. C. Greubel_, Apr 09 2018
%Y A113011 Cf. A019774, A113012, A113013.
%K A113011 nonn,cons
%O A113011 1,2
%A A113011 _Eric W. Weisstein_, following a suggestion of Grover W. Hughes, Oct 09 2005
%E A113011 Simpler definition from _T. D. Noe_, Oct 09 2005
%E A113011 Euler reference from _David L. Harden_, Oct 09 2005