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A202543
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Decimal expansion of the number x satisfying e^(x/2) - e^(-x/2) = 1.
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7
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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
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0,1
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COMMENTS
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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
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LINKS
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FORMULA
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Equals Sum_{k>=0} (-1)^k*binomial(2*k,k)/((2*k+1)*16^k). (End)
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EXAMPLE
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0.9624236501192068949955178268487368462703686...
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MATHEMATICA
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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[ FindRoot[ Exp[x/2] == 1 + Exp[-x/2] , {x, 0}, WorkingPrecision -> 128][[1, 2]]][[1]] (* Robert G. Wilson v, Jun 13 2018 *)
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PROG
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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