A053004 Decimal expansion of AGM(1,sqrt(2)).

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

Offset

1,3

Comments

AGM(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

George Boros and Victor H. Moll, Irresistible integrals, Cambridge University Press (2006), p. 195.

J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, page 5.

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 6.1, p. 420.

J. R. Goldman, The Queen of Mathematics, 1998, p. 92.

Links

Harry J. Smith, Table of n, a(n) for n = 1..20000

Eric Weisstein's World of Mathematics, Gauss's Constant.

Index entries for transcendental numbers.

Formula

Equals Pi/(2*A085565). - Nathaniel Johnston, May 26 2011

Equals Integral_{x=0..Pi/2} sqrt(sin(x)) or Integral_{x=0..1} sqrt(x/(1-x^2)). - Jean-François Alcover, Apr 29 2013 [cf. Boros & Moll p. 195]

Equals Product_{n>=1} (1+1/A033566(n)) and also 2*AGM(1, i)/(1+i) where i is the imaginary unit. - Dimitris Valianatos, Oct 03 2016

Conjecturally equals 1/( Sum_{n = -infinity..infinity} exp(-Pi*(n+1/2)^2 ) )^2. Cf. A096427. - Peter Bala, Jun 10 2019

From Amiram Eldar, Aug 26 2020: (Start)

Equals 2 * A076390.

Equals Integral_{x=0..Pi} sin(x)^2/sqrt(1 + sin(x)^2) dx. (End)

Equals sqrt(2/Pi)*Gamma(3/4)^2 = Integral_{x = 0..1} 1/(1 - x^2)^(1/4) dx = Pi/Integral_{x = 0..1} 1/(1 - x^2)^(3/4) dx. - Peter Bala, Jan 05 2022

From Peter Bala, Mar 02 2022: (Start)

Equals 2*Integral_{x = 0..1} x^2/sqrt(1 - x^4) dx.

Equals 1 - Integral_{x = 0..1} (sqrt(1 - x^4) - 1)/x^2 dx.

Equals hypergeom([-1/2, -1/4], [3/4], 1) = 1 + Sum_{n >= 0} 1/(4*n + 3)*Catalan(n)*(1/2^(2*n+1)). Cf. A096427. (End)

Example

1.19814023473559220743992249228...

Maple

evalf(GaussAGM(1, sqrt(2)), 144);  # Alois P. Heinz, Jul 05 2023

Mathematica

RealDigits[ N[ ArithmeticGeometricMean[1, Sqrt[2]], 105]][[1]] (* Jean-François Alcover, Jan 30 2012 *)
RealDigits[N[(1+Sqrt[2])Pi/(4EllipticK[17-12Sqrt[2]]), 105]][[1]] (* Jean-François Alcover, Jun 02 2019 *)

Prog

(PARI) default(realprecision, 20080); x=agm(1, sqrt(2)); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b053004.txt", n, " ", d)) \\ Harry J. Smith, Apr 20 2009
(PARI) 2*real(agm(1, I)/(1+I)) \\ Michel Marcus, Jul 26 2018
(Python)
from mpmath import mp, agm, sqrt
mp.dps=106
print([int(z) for z in list(str(agm(1, sqrt(2))).replace('.', '')[:-1])]) # Indranil Ghosh, Jul 11 2017

Crossrefs

Cf. A014549, A053002, A053003, A076390, A085565, A096427.

Sequence in context: A198920 A354593 A244115 * A019888 A154975 A199472

Adjacent sequences: A053001 A053002 A053003 * A053005 A053006 A053007

Keyword

nonn,cons,nice,easy

Author

N. J. A. Sloane, Feb 21 2000

Extensions

More terms from James A. Sellers, Feb 22 2000

Status

approved