A129187 - OEIS

%I #31 Nov 21 2024 08:05:50

%S 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,

%T 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,

%U 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

%N Decimal expansion of arcsinh(1/3).

%C 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

%H Muniru A Asiru, <a href="/A129187/b129187.txt">Table of n, a(n) for n = 0..2000</a>

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>

%F Equals log((1 + sqrt(10))/3). - _Jianing Song_, Jul 12 2018

%F Equals arccoth(sqrt(10)). - _Amiram Eldar_, Feb 09 2024

%e 0.32745015023725844332253525998825812770052452899076745127562...

%t RealDigits[ArcSinh[1/3], 10, 111][[1]] (* _Robert G. Wilson v_, Jul 23 2018 *)

%o (PARI) asinh(1/3) \\ _Charles R Greathouse IV_, Mar 25 2014

%Y Cf. A002390, A129200, A129269, A244644.

%K nonn,cons,changed

%O 0,1

%A _N. J. A. Sloane_, Jul 27 2008