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A195621
Decimal expansion of arccsc(4).
4
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, 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, 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
OFFSET
0,1
COMMENTS
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
LINKS
FORMULA
Equals arccos(sqrt(15)/4) = arctan(1/sqrt(15)). - Amiram Eldar, Jul 11 2023
EXAMPLE
arccsc(4) = arcsin(1/4) = 0.25268025514207865...
MATHEMATICA
(See A195628.)
RealDigits[ ArcCsc@4, 10, 111][[1]] (* Robert G. Wilson v, Jul 23 2018 *)
PROG
(PARI) asin(1/4) \\ Michel Marcus, Jul 12 2018
(Magma) SetDefaultRealField(RealField(100)); Arcsin(1/4); // G. C. Greubel, Nov 11 2018
CROSSREFS
Sequence in context: A121311 A242403 A179015 * A267090 A067948 A142148
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Sep 23 2011
STATUS
approved