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

Author

Clark Kimberling, Sep 23 2011

Keywords

• nonn
• cons

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

Examples

			arccsc(4) = arcsin(1/4) = 0.25268025514207865...
		

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.

Crossrefs

Cf. A195628, A244644.

Programs

Magma

SetDefaultRealField(RealField(100)); Arcsin(1/4); // G. C. Greubel, Nov 11 2018

Mathematica

(See A195628.)
RealDigits[ ArcCsc@4, 10, 111][[1]] (* Robert G. Wilson v, Jul 23 2018 *)

PARI

asin(1/4) \\ Michel Marcus, Jul 12 2018

Formula

Equals arccos(sqrt(15)/4) = arctan(1/sqrt(15)). - Amiram Eldar, Jul 11 2023