A129200 Decimal expansion of arcsinh(1/4).

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

Offset: 0

Author

N. J. A. Sloane, Jul 27 2008

Keywords

• nonn
• cons

Comments

Archimedes's-like scheme: set p(0) = 1/sqrt(17), 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

Examples

			.24746646154726345294478154978835928925376690309856769646911...
		

Links

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

Programs

Magma

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

Mathematica

RealDigits[ArcSinh[1/4], 10, 111][[1]] (* Robert G. Wilson v, Jul 23 2018 *)

PARI

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

Formula

Equals log((1 + sqrt(17))/4). - Jianing Song, Jul 12 2018