OFFSET
0,1
COMMENTS
Archimedes's-like scheme: set p(0) = 1/sqrt(15), q(0) = 1/4; p(n+1) = 2*p(n)*q(n)/(p(n)+q(n)) (harmonic mean, 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
Muniru A Asiru, Table of n, a(n) for n = 0..2000
FORMULA
Equals arccos(sqrt(15)/4) = arctan(1/sqrt(15)). - Amiram Eldar, Jul 11 2023
EXAMPLE
arccsc(4) = arcsin(1/4) = 0.25268025514207865...
MATHEMATICA
(See A195628.)
RealDigits[ ArcCsc@4, 10, 111][[1]] (* Robert G. Wilson v, Jul 23 2018 *)
PROG
(PARI) asin(1/4) \\ Michel Marcus, Jul 12 2018
(Magma) SetDefaultRealField(RealField(100)); Arcsin(1/4); // G. C. Greubel, Nov 11 2018
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Sep 23 2011
STATUS
approved