

A014549


Decimal expansion of 1 / M(1,sqrt(2)) (Gauss's constant).


11



8, 3, 4, 6, 2, 6, 8, 4, 1, 6, 7, 4, 0, 7, 3, 1, 8, 6, 2, 8, 1, 4, 2, 9, 7, 3, 2, 7, 9, 9, 0, 4, 6, 8, 0, 8, 9, 9, 3, 9, 9, 3, 0, 1, 3, 4, 9, 0, 3, 4, 7, 0, 0, 2, 4, 4, 9, 8, 2, 7, 3, 7, 0, 1, 0, 3, 6, 8, 1, 9, 9, 2, 7, 0, 9, 5, 2, 6, 4, 1, 1, 8, 6, 9, 6, 9, 1, 1, 6, 0, 3, 5, 1, 2, 7, 5, 3, 2, 4, 1, 2, 9, 0, 6, 7, 8, 5
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OFFSET

0,1


COMMENTS

On May 30, 1799, Gauss discovered that this number is also equal to (2/Pi)*Integral_{t=0..1} 1/sqrt(1t^4).
M(a,b) is the limit of the arithmeticgeometric mean iteration applied repeatedly starting with a and b: a_0 = a, b_0 = b, a_{n+1} = (a_n + b_n)/2, b_{n+1} = sqrt(a_n*b_n).


REFERENCES

J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, page 5.
J. R. Goldman, The Queen of Mathematics, 1998, p. 92.


LINKS

Harry J. Smith, Table of n, a(n) for n = 0..20000
Eric Weisstein's World of Mathematics, Gauss's Constant
Eric Weisstein's World of Mathematics, ArithmeticGeometric Mean


FORMULA

Equals (lim {k > inf} p(k))/(1+i) and (lim {k > inf} q(k))/(1+i), where i is the imaginary unit, p(0) = 1, q(0) = i, p(k+1) = 2*p(k)*q(k)/(p(k)+q(k)) and q(k+1) = sqrt(p(k)*q(k)) for k >= 0.  A.H.M. Smeets, Jul 26 2018


EXAMPLE

0.8346268416740731862814297327990468...


MATHEMATICA

RealDigits[Gamma[1/4]^2/(2*Pi^(3/2)*Sqrt[2]), 10, 105][[1]] (* or: *)
RealDigits[1/ArithmeticGeometricMean[1, Sqrt[2]], 10, 105][[1]] (* JeanFrançois Alcover, Dec 13 2011, updated Nov 11 2016, after Eric W. Weisstein *)


PROG

(PARI) default(realprecision, 20080); x=10*agm(1, sqrt(2))^1; for (n=0, 20000, d=floor(x); x=(xd)*10; write("b014549.txt", n, " ", d)); \\ Harry J. Smith, Apr 20 2009
(PARI) 1/agm(sqrt(2), 1) \\ Charles R Greathouse IV, Feb 04 2015
(PARI) sqrt(Pi/2)/gamma(3/4)^2 \\ Charles R Greathouse IV, Feb 04 2015
(Python)
from mpmath import *
mp.dps=105
print map(int, list(str(1/agm(sqrt(2)))[2:1])) # Indranil Ghosh, Jul 11 2017
(MAGMA) SetDefaultRealField(RealField(100)); R:= RealField(); Sqrt(Pi(R)/2)/Gamma(3/4)^2; // G. C. Greubel, Aug 17 2018


CROSSREFS

Cf. A053002, A053003, A053004, A244644.
Sequence in context: A222232 A091895 A111436 * A021549 A013665 A209059
Adjacent sequences: A014546 A014547 A014548 * A014550 A014551 A014552


KEYWORD

nonn,cons,nice


AUTHOR

Eric W. Weisstein, N. J. A. Sloane


EXTENSIONS

Extended to 105 terms by JeanFrançois Alcover, Dec 13 2011
a(104) corrected by Andrew Howroyd, Feb 23 2018


STATUS

approved



