A202541 Decimal expansion of the number x satisfying e^(2x) - e^(-2x) = 1.

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, 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, 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

Offset: 0

Author

Clark Kimberling, Dec 21 2011

Keywords

• nonn
• cons

Comments

See A202537 for a guide to related sequences. The Mathematica program includes a graph.

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

Examples

			0.24060591252980172374887945671218421156759216719...
		

Links

• R. S. Melham and A. G. Shannon, Inverse trigonometric and hyperbolic summation formulas involving generalized Fibonacci numbers, Fibonacci Quarterly, Vol. 33, No. 1 (1995), pp. 32-40.
• D. Zagier, Algebraic numbers close to both 0 and 1, Mathematics of Computation, Vol. 61, No. 203 (1993), pp. 485-491.
• Index entries for transcendental numbers

Crossrefs

Cf. A001622, A002390, A202537.

Programs

Mathematica

u = 2; v = 2;
f[x_] := E^(u*x) - E^(-v*x); g[x_] := 1
Plot[{f[x], g[x]}, {x, -2, 2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, .2, .3}, WorkingPrecision -> 110]
RealDigits[r]   (* A202541 *)
RealDigits[ Log[ (1+Sqrt[5])/2 ] / 2, 10, 99] // First (* Jean-François Alcover, Feb 27 2013 *)
RealDigits[ FindRoot[ Exp[2x] - Exp[-2x] == 1, {x, 1}, WorkingPrecision -> 128][[1, 2]], 10, 111][[1]] (* Robert G. Wilson v, Jul 23 2018 *)

PARI

asinh(1/2)/2 \\ Michel Marcus, Jul 12 2018

Formula

Equals (1/2)*arcsinh(1/2) or (1/2)*log(phi), phi being the golden ratio. - A.H.M. Smeets, Jul 12 2018

Equals Sum_{k>=1} (-1)^(k+1) * arctanh(1/Fibonacci(3*k)^2) (Melham and Shannon, 1995). - Amiram Eldar, Oct 04 2021

Equals A002390/2. - Alois P. Heinz, Jul 14 2022

Equals arctanh(sqrt(5)-2). - Amiram Eldar, Feb 09 2024