A195621 Decimal expansion of arccsc(4).

2, 5, 2, 6, 8, 0, 2, 5, 5, 1, 4, 2, 0, 7, 8, 6, 5, 3, 4, 8, 5, 6, 5, 7, 4, 3, 6, 9, 9, 3, 7, 1, 0, 9, 7, 2, 2, 5, 2, 1, 9, 3, 7, 3, 3, 0, 9, 6, 8, 3, 8, 1, 9, 3, 6, 3, 3, 9, 2, 3, 7, 7, 8, 7, 4, 0, 5, 7, 5, 0, 6, 0, 4, 8, 1, 0, 2, 1, 2, 2, 2, 4, 1, 1, 7, 4, 8, 7, 4, 2, 2, 2, 8, 0, 1, 4, 6, 0, 1, 6

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

References

Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 35, page 338.

Links

Muniru A Asiru, Table of n, a(n) for n = 0..2000

Index entries for transcendental numbers.

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

Cf. A195628, A244644.

Sequence in context: A121311 A242403 A179015 * A267090 A067948 A142148

Adjacent sequences: A195618 A195619 A195620 * A195622 A195623 A195624

Keyword

nonn,cons,changed

Author

Clark Kimberling, Sep 23 2011

Status

approved