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%I #54 Jul 12 2021 15:59:13
%S 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,
%T 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,
%U 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
%N Decimal expansion of the number x satisfying e^(x/2) - e^(-x/2) = 1.
%C See A202537 for a guide to related sequences. The Mathematica program includes a graph.
%C 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
%C 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
%H W. Gawronski, L. L. Littlejohn, and T. Neuschel, <a href="http://arxiv.org/abs/1308.6803">Asymptotics of Stirling and Chebyshev-Stirling numbers of the second kind</a>, arXiv:1308.6803 [math.CO], 2013.
%H W. Gawronski, L. L. Littlejohn, and T. Neuschel, <a href="http://dx.doi.org/10.1111/sapm.12037">Asymptotics of Stirling and Chebyshev-Stirling numbers of the second kind</a>, Studies in Applied Mathematics by MIT 133 (2014), 1-17.
%H Dan Lascu, <a href="https://doi.org/10.1016/j.jnt.2012.12.007">A Gauss-Kuzmintype problem for a family of continued fraction expansions</a>, Journal of Number Theory 133 (2013), 2153-2181.
%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F Equals 2*A002390. - _A.H.M. Smeets_, Jun 06 2018
%F From _Amiram Eldar_, Aug 21 2020: (Start)
%F Equals log(A104457) = log(1 + A001622).
%F Equals 2*arcsinh(1/2). [corrected by _Georg Fischer_, Jul 12 2021]
%F Equals Sum_{k>=0} (-1)^k*binomial(2*k,k)/((2*k+1)*16^k). (End)
%e 0.9624236501192068949955178268487368462703686...
%t u = 1/2; v = 1/2;
%t f[x_] := E^(u*x) - E^(-v*x); g[x_] := 1
%t Plot[{f[x], g[x]}, {x, 0, 2}, {AxesOrigin -> {0, 0}}]
%t r = x /. FindRoot[f[x] == g[x], {x, .9, 1}, WorkingPrecision -> 110]
%t RealDigits[r] (* A202543 *)
%t RealDigits[ Log[ (3+Sqrt[5])/2], 10, 99] // First (* _Jean-François Alcover_, Feb 27 2013 *)
%t RealDigits[ FindRoot[ Exp[x/2] == 1 + Exp[-x/2] , {x, 0}, WorkingPrecision -> 128][[1, 2]]][[1]] (* _Robert G. Wilson v_, Jun 13 2018 *)
%o (PARI) 2*asinh(1/2) \\ _Michel Marcus_, Jun 24 2018, after A002390
%Y Cf. A001622, A002390, A104457, A202537, A202543.
%K nonn,cons
%O 0,1
%A _Clark Kimberling_, Dec 21 2011
%E Typo in name fixed by _Jean-François Alcover_, Feb 27 2013
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