A113011 Decimal expansion of 1/(sqrt(e) - 1).
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, 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, 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
Offset: 1
Examples
1.54149408253679828413110344447251463834045923684188210947413695663...
Links
- G. C. Greubel, Table of n, a(n) for n = 1..20000
- Itay Beit-Halachmi and Ido Kaminer, The Ramanujan Library -- Automated Discovery on the Hypergraph of Integer Relations, arXiv:2412.12361 [cs.AI], 2024. See p. 6.
- Leonhard Euler, On the formation of continued fractions, arXiv:math/0508227 [math.HO], 2005, see p. 14.
- Michel Waldschmidt, Continued fractions, Ecole de recherche CIMPA-Oujda, Théorie des Nombres et ses Applications, 18 - 29 mai 2015: Oujda (Maroc).
- Eric Weisstein's World of Mathematics, Continued Fraction.
- Index entries for transcendental numbers.
Programs
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Magma
1/(Sqrt(Exp(1)) - 1); // G. C. Greubel, Apr 09 2018
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Mathematica
First@ RealDigits[ 1 / (Exp[1/2] - 1), 10, 111] (* Robert G. Wilson v, Jul 01 2007 *) 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 *) Rest[realDigitsRecip[Sqrt[E]-1]] (* The realDigitsRecip program is at A021200 *) (* Harvey P. Dale, Nov 04 2024 *)
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PARI
1/(sqrt(exp(1)) - 1) \\ G. C. Greubel, Apr 09 2018
Formula
Equals Integral_{x = 0..oo} floor(2*x)*exp(-x) dx. - Peter Bala, Oct 09 2013
Equals 3/2 + Sum_{k>=0} tanh(1/2^(k+3))/2^(k+2). - Antonio Graciá Llorente, Jan 21 2024
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
Equals 1/(A019774 - 1). - Hugo Pfoertner, Dec 27 2024
Extensions
Simpler definition from T. D. Noe, Oct 09 2005
Euler reference from David L. Harden, Oct 09 2005
Comments