

A202543


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


7



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
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OFFSET

0,1


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 ChebyshevStirling 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


LINKS

Table of n, a(n) for n=0..98.
W. Gawronski, L. L. Littlejohn, and T. Neuschel, Asymptotics of Stirling and ChebyshevStirling numbers of the second kind, arXiv:1308.6803 [math.CO], 2013.
W. Gawronski, L. L. Littlejohn, and T. Neuschel, Asymptotics of Stirling and ChebyshevStirling numbers of the second kind, Studies in Applied Mathematics by MIT 133 (2014), 117.
Dan Lascu, A GaussKuzmintype problem for a family of continued fraction expansions, Journal of Number Theory 133 (2013), 21532181.
Index entries for transcendental numbers


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)


EXAMPLE

0.9624236501192068949955178268487368462703686...


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 (* JeanFrançois 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 *)


PROG

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


CROSSREFS

Cf. A001622, A002390, A104457, A202537, A202543.
Sequence in context: A154899 A335563 A011219 * A188528 A243257 A194182
Adjacent sequences: A202540 A202541 A202542 * A202544 A202545 A202546


KEYWORD

nonn,cons


AUTHOR

Clark Kimberling, Dec 21 2011


EXTENSIONS

Typo in name fixed by JeanFrançois Alcover, Feb 27 2013


STATUS

approved



