%I
%S 2,4,0,6,0,5,9,1,2,5,2,9,8,0,1,7,2,3,7,4,8,8,7,9,4,5,6,7,1,2,1,8,4,2,
%T 1,1,5,6,7,5,9,2,1,6,7,1,9,2,8,3,0,2,5,9,8,3,0,5,0,9,0,8,4,4,2,0,0,8,
%U 1,9,3,3,8,0,4,1,1,0,8,8,7,2,0,6,0,0,4,7,1,4,5,6,1,3,6,1,7,3,7
%N Decimal expansion of the number x satisfying e^(2x)  e^(2x) = 1.
%C See A202537 for a guide to related sequences. The Mathematica program includes a graph.
%C Archimedes'slike scheme: set p(0) = 1/(2*sqrt(5)), q(0) = 1/4; p(n+1) = 2*p(n)*q(n)/(p(n)+q(n)) (arithmetic mean of reciprocals, i.e., 1/p(n+1) = (1/p(n) + 1/q(n))/2), q(n+1) = sqrt(p(n+1)*q(n)) (geometric mean, i.e., log(q(n+1)) = (log(p(n+1)) + log(q(n)))/2), for n >= 0. The error of p(n) and q(n) decreases by a factor of approximately 4 each iteration, i.e., approximately 2 bits are gained by each iteration. Set r(n) = (2*q(n) + p(n))/3, the error decreases by a factor of approximately 16 for each iteration, i.e., approximately 4 bits are gained by each iteration. For a similar scheme see also A244644.  _A.H.M. Smeets_, Jul 12 2018
%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F Equals (1/2)*arcsinh(1/2) or (1/2)*log(phi), phi being the golden ratio.  _A.H.M. Smeets_, Jul 12 2018
%e x = 0.24060591252980172374887945671218421156759216719...
%t u = 2; v = 2;
%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] (* A202541 *)
%t RealDigits[ Log[ (1+Sqrt[5])/2 ] / 2, 10, 99] // First (* _JeanFrançois Alcover_, Feb 27 2013 *)
%t RealDigits[ FindRoot[ Exp[2x]  Exp[2x] == 1, {x, 1}, WorkingPrecision > 128][[1, 2]], 10, 111][[1]] (* _Robert G. Wilson v_, Jul 23 2018 *)
%o (PARI) asinh(1/2)/2 \\ _Michel Marcus_, Jul 12 2018
%Y Cf. A001622, A202537.
%K nonn,cons
%O 0,1
%A _Clark Kimberling_, Dec 21 2011
