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%I #41 Feb 09 2024 03:26:15
%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's-like 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 R. S. Melham and A. G. Shannon, <a href="https://www.fq.math.ca/Scanned/33-1/melham2.pdf">Inverse trigonometric and hyperbolic summation formulas involving generalized Fibonacci numbers</a>, Fibonacci Quarterly, Vol. 33, No. 1 (1995), pp. 32-40.
%H D. Zagier, <a href="https://people.mpim-bonn.mpg.de/zagier/files/doi/10.2307/2152970/fulltext.pdf">Algebraic numbers close to both 0 and 1</a>, Mathematics of Computation, Vol. 61, No. 203 (1993), pp. 485-491.
%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
%F Equals Sum_{k>=1} (-1)^(k+1) * arctanh(1/Fibonacci(3*k)^2) (Melham and Shannon, 1995). - _Amiram Eldar_, Oct 04 2021
%F Equals A002390/2. - _Alois P. Heinz_, Jul 14 2022
%F Equals arctanh(sqrt(5)-2). - _Amiram Eldar_, Feb 09 2024
%e 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 (* _Jean-François 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, A002390, A202537.
%K nonn,cons
%O 0,1
%A _Clark Kimberling_, Dec 21 2011
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