A195621 - OEIS

%I #51 Jan 03 2025 11:48:31

%S 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,

%T 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,

%U 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

%N Decimal expansion of arccsc(4).

%C 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

%D Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 35, page 338.

%H Muniru A Asiru, <a href="/A195621/b195621.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 arccos(sqrt(15)/4) = arctan(1/sqrt(15)). - _Amiram Eldar_, Jul 11 2023

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

%t (See A195628.)

%t RealDigits[ ArcCsc@4, 10, 111][[1]] (* _Robert G. Wilson v_, Jul 23 2018 *)

%o (PARI) asin(1/4) \\ _Michel Marcus_, Jul 12 2018

%o (Magma) SetDefaultRealField(RealField(100)); Arcsin(1/4); // _G. C. Greubel_, Nov 11 2018

%Y Cf. A195628, A244644.

%K nonn,cons,changed

%O 0,1

%A _Clark Kimberling_, Sep 23 2011