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%I #13 Jul 14 2022 12:40:54
%S 2,8,1,1,9,9,5,7,4,3,2,2,9,6,1,8,4,6,5,1,2,0,5,0,7,6,4,0,6,7,8,7,8,2,
%T 9,9,7,9,2,0,2,3,2,2,5,7,4,4,0,6,6,4,6,2,6,7,5,7,3,0,3,3,4,3,1,8,0,3,
%U 8,4,5,3,0,6,2,1,2,0,8,9,1,3,2,2,9,8,7,7,0,7,4,7,5,4,9,4,0,5,4
%N Decimal expansion of the number x satisfying e^(2x)-e^(-x)=1.
%C See A202537 for a guide to related sequences. The Mathematica program includes a graph.
%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F Equals log((v^2+12)/(6*v)) with v = (108+12*sqrt(69))^(1/3). - _Alois P. Heinz_, Jul 14 2022
%e x=0.281199574322961846512050764067878299792023...
%t u = 2; v = 1;
%t f[x_] := E^(u*x) - E^(-v*x); g[x_] := 1
%t Plot[{f[x], g[x]}, {x, -2, 2}, {AxesOrigin -> {0, 0}}]
%t r = x /. FindRoot[f[x] == g[x], {x, .2, .3}, WorkingPrecision -> 110]
%t RealDigits[r] (* A202539 *)
%t RealDigits[ Log[ Root[#^3 - # - 1&, 1]], 10, 99] // First (* _Jean-François Alcover_, Feb 27 2013 *)
%o (PARI) log(polrootsreal(x^3-x-1)[1]) \\ _Charles R Greathouse IV_, May 15 2019
%Y Cf. A202537.
%K nonn,cons
%O 0,1
%A _Clark Kimberling_, Dec 21 2011
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