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A014549 Decimal expansion of 1 / M(1,sqrt(2)) (Gauss's constant). 10
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, 8 (list; constant; graph; refs; listen; history; text; internal format)
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(1-t^4).

M(a,b) is the limit of the arithmetic-geometric 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, Arithmetic-Geometric Mean

EXAMPLE

0.8346268416740731862814297327990468...

MATHEMATICA

RealDigits[ N[ Gamma[1/4]^2/(2*Pi^(3/2)*Sqrt[2]), 105]][[1]]  (* Jean-François Alcover, Dec 13 2011, after Eric W. Weisstein *)

PROG

(PARI) default(realprecision, 20080); x=10*agm(1, sqrt(2))^-1; for (n=0, 20000, d=floor(x); x=(x-d)*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

CROSSREFS

Cf. A053002, A053003, A053004.

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 Jean-François Alcover, Dec 13 2011

STATUS

approved

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Last modified February 11 08:09 EST 2016. Contains 268163 sequences.