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A129200 Decimal expansion of arcsinh(1/4). 3
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

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

LINKS

Muniru A Asiru, Table of n, a(n) for n = 0..2000

FORMULA

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

EXAMPLE

.24746646154726345294478154978835928925376690309856769646911...

MATHEMATICA

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

PROG

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

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

CROSSREFS

Sequence in context: A239324 A114375 A198503 * A074958 A222407 A230724

Adjacent sequences:  A129197 A129198 A129199 * A129201 A129202 A129203

KEYWORD

nonn,cons

AUTHOR

N. J. A. Sloane, Jul 27 2008

STATUS

approved

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Last modified October 15 00:14 EDT 2019. Contains 328025 sequences. (Running on oeis4.)