OFFSET
0,1
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
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.
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
EXAMPLE
0.24060591252980172374887945671218421156759216719...
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 *)
PROG
(PARI) asinh(1/2)/2 \\ Michel Marcus, Jul 12 2018
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Dec 21 2011
STATUS
approved