

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'slike 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



