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%I #27 Sep 08 2022 08:45:30
%S 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,
%T 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,
%U 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
%N Decimal expansion of arcsinh(1/4).
%C 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
%H Muniru A Asiru, <a href="/A129200/b129200.txt">Table of n, a(n) for n = 0..2000</a>
%F Equals log((1 + sqrt(17))/4). - _Jianing Song_, Jul 12 2018
%e .24746646154726345294478154978835928925376690309856769646911...
%t RealDigits[ArcSinh[1/4], 10, 111][[1]] (* _Robert G. Wilson v_, Jul 23 2018 *)
%o (PARI) asinh(1/4) \\ _Michel Marcus_, Jul 12 2018
%o (Magma) SetDefaultRealField(RealField(100)); Argsinh(1/4); // _G. C. Greubel_, Nov 11 2018
%K nonn,cons
%O 0,1
%A _N. J. A. Sloane_, Jul 27 2008
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