A129187 Decimal expansion of arcsinh(1/3).

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

Offset: 0

Author

N. J. A. Sloane, Jul 27 2008

Keywords

• nonn
• cons

Comments

Archimedes's-like scheme: set p(0) = 1/sqrt(10), q(0) = 1/3; 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

			0.32745015023725844332253525998825812770052452899076745127562...
		

Links

• Muniru A Asiru, Table of n, a(n) for n = 0..2000
• Index entries for transcendental numbers

Crossrefs

Cf. A002390, A129200, A129269, A244644.

Programs

Mathematica

RealDigits[ArcSinh[1/3], 10, 111][[1]] (* Robert G. Wilson v, Jul 23 2018 *)

PARI

asinh(1/3) \\ Charles R Greathouse IV, Mar 25 2014

Formula

Equals log((1 + sqrt(10))/3). - Jianing Song, Jul 12 2018

Equals arccoth(sqrt(10)). - Amiram Eldar, Feb 09 2024