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

Author

N. J. A. Sloane, Feb 21 2000

Keywords

• nonn
• cons
• nice
• easy

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).

Examples

			1.19814023473559220743992249228...
		

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.

Crossrefs

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

Programs

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 *)

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

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)

Extensions

More terms from James Sellers, Feb 22 2000