A202543 Decimal expansion of the number x satisfying e^(x/2) - e^(-x/2) = 1.

9, 6, 2, 4, 2, 3, 6, 5, 0, 1, 1, 9, 2, 0, 6, 8, 9, 4, 9, 9, 5, 5, 1, 7, 8, 2, 6, 8, 4, 8, 7, 3, 6, 8, 4, 6, 2, 7, 0, 3, 6, 8, 6, 6, 8, 7, 7, 1, 3, 2, 1, 0, 3, 9, 3, 2, 2, 0, 3, 6, 3, 3, 7, 6, 8, 0, 3, 2, 7, 7, 3, 5, 2, 1, 6, 4, 4, 3, 5, 4, 8, 8, 2, 4, 0, 1, 8, 8, 5, 8, 2, 4, 5, 4, 4, 6, 9, 4, 9

Offset: 0

Author

Clark Kimberling, Dec 21 2011

Keywords

• nonn
• cons

Comments

See A202537 for a guide to related sequences. The Mathematica program includes a graph.

W. Gawronski et al. in their paper - see ref. below - obtained the asymptotics for the Chebyshev-Stirling numbers. In the algebraic description of the respective "asymptotic coefficients" the number x = 2*log phi, where phi is the golden section, play the central role. - Roman Witula, Feb 02 2015

Also two times the Lévy measure for the continued fraction of the golden section, i.e., A202543/log(2) is the mean number of bits gained from the next convergent of the continued fraction representation. (See also Dan Lascu in links.) - A.H.M. Smeets, Jun 06 2018

Examples

			0.9624236501192068949955178268487368462703686...
		

Links

• W. Gawronski, L. L. Littlejohn, and T. Neuschel, Asymptotics of Stirling and Chebyshev-Stirling numbers of the second kind, arXiv:1308.6803 [math.CO], 2013.
• W. Gawronski, L. L. Littlejohn, and T. Neuschel, Asymptotics of Stirling and Chebyshev-Stirling numbers of the second kind, Studies in Applied Mathematics by MIT 133 (2014), 1-17.
• Dan Lascu, A Gauss-Kuzmintype problem for a family of continued fraction expansions, Journal of Number Theory 133 (2013), 2153-2181.
• Index entries for transcendental numbers

Crossrefs

Cf. A001622, A002390, A104457, A202537, A202543.

Programs

Mathematica

u = 1/2; v = 1/2;
f[x_] := E^(u*x) - E^(-v*x); g[x_] := 1
Plot[{f[x], g[x]}, {x, 0, 2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, .9, 1}, WorkingPrecision -> 110]
RealDigits[r]    (* A202543 *)
RealDigits[ Log[ (3+Sqrt[5])/2], 10, 99] // First (* Jean-François Alcover, Feb 27 2013 *)
RealDigits[ FindRoot[ Exp[x/2] == 1 +  Exp[-x/2] , {x, 0}, WorkingPrecision -> 128][[1, 2]]][[1]] (* Robert G. Wilson v, Jun 13 2018 *)

PARI

2*asinh(1/2) \\ Michel Marcus, Jun 24 2018, after A002390

Formula

Equals 2*A002390. - A.H.M. Smeets, Jun 06 2018

From Amiram Eldar, Aug 21 2020: (Start)

Equals log(A104457) = log(1 + A001622).

Equals 2*arcsinh(1/2). [corrected by Georg Fischer, Jul 12 2021]

Equals Sum_{k>=0} (-1)^k*binomial(2*k,k)/((2*k+1)*16^k). (End)

Equals Pi*i + Sum_{k>=0} arctanh(phi^(2^k))/2^k, with phi = A001622 and i = sqrt(-1). - Antonio Graciá Llorente, Feb 13 2025

Extensions

Typo in name fixed by Jean-François Alcover, Feb 27 2013