A053002 -id:A053002 - cp's OEIS frontend

A014549 Decimal expansion of 1 / M(1,sqrt(2)) (Gauss's constant).

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

Offset: 0

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

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).

			0.8346268416740731862814297327990468...

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, Sections 1.5.4 and 6.1, pp. 34, 420.
J. R. Goldman, The Queen of Mathematics, 1998, p. 92.

Harry J. Smith, Table of n, a(n) for n = 0..20000
Markus Faulhuber, An application of hypergeometric functions to heat kernels on rectangular and hexagonal tori and a "Weltkonstante"-or-how Ramanujan split temperatures, The Ramanujan Journal volume 54, pages 1-27 (2021). See pp. 4 and 24.
Markus Faulhuber, Anupam Gumber, and Irina Shafkulovska, The AGM of Gauss, Ramanujan's corresponding theory, and spectral bounds of self-adjoint operators, arXiv:2209.04202 [math.CA], 2022, p. 2.
Alessandro Languasco and Pieter Moree, Euler constants from primes in arithmetic progression, arXiv:2406.16547 [math.NT], 2024. See p. 10.
Eric Weisstein's World of Mathematics, Gauss's Constant.
Eric Weisstein's World of Mathematics, Arithmetic-Geometric Mean.
Index entries for transcendental numbers.

Cf. A053002, A053003, A053004, A062539, A093341, A244644, A248557.

SetDefaultRealField(RealField(100)); R:= RealField(); Sqrt(Pi(R)/2)/Gamma(3/4)^2; // G. C. Greubel, Aug 17 2018

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

RealDigits[Gamma[1/4]^2/(2*Pi^(3/2)*Sqrt[2]), 10, 105][[1]] (* or: *)
RealDigits[1/ArithmeticGeometricMean[1, Sqrt[2]], 10, 105][[1]] (* Jean-François Alcover, Dec 13 2011, updated Nov 11 2016, after Eric W. Weisstein *)
First[RealDigits[N[EllipticTheta[4, Exp[-Pi]]^2, 90]]] (* Stefano Spezia, Sep 29 2022 *)

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

1/agm(sqrt(2),1) \\ Charles R Greathouse IV, Feb 04 2015

sqrt(Pi/2)/gamma(3/4)^2 \\ Charles R Greathouse IV, Feb 04 2015

from mpmath import mp, agm, sqrt
mp.dps=105
print([int(z) for z in list(str(1/agm(sqrt(2)))[2:-1])]) # Indranil Ghosh, Jul 11 2017

Equals (lim_{k->oo} p(k))/(1+i) and (lim_{k->oo} 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
Equals the infinite quotient product (3/4)*(6/5)*(7/8)*(10/9)*(11/12)*(14/13)*(15/16)*... . - James Maclachlan, Jul 28 2019
Equals (9/15)*hypergeom([1/2, 3/4], [9/4], 1). - Peter Bala, Mar 03 2022
Equals A062539 / Pi. - Amiram Eldar, May 04 2022
From Stefano Spezia, Sep 29 2022: (Start)
Equals theta4(exp(-Pi))^2.
Equals sqrt(2)*A093341/Pi. (End)
Equals Sum_{k>=0} (-1)^k * binomial(2*k,k)^2/16^k. - Amiram Eldar, Jul 04 2023
From Gerry Martens, Jul 31 2023: (Start)
Equals 2*Gamma(5/4)/(sqrt(Pi)*Gamma(3/4)).
Equals hypergeom([1/4, -2/4], [1], 1). (End)
Equals A248557^2. - Hugo Pfoertner, Jun 28 2024

Extended to 105 terms by Jean-François Alcover, Dec 13 2011

a(104) corrected by Andrew Howroyd, Feb 23 2018

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

Original entry on oeis.org

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

N. J. A. Sloane, Feb 21 2000

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).

			1.19814023473559220743992249228...

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.

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.

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

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

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 *)

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

2*real(agm(1, I)/(1+I)) \\ Michel Marcus, Jul 26 2018

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

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)

More terms from James Sellers, Feb 22 2000

A053003 Continued fraction for M(1,sqrt(2)).

Original entry on oeis.org

1, 5, 21, 3, 4, 14, 1, 1, 1, 1, 1, 3, 1, 15, 1, 3, 8, 36, 1, 2, 5, 2, 1, 1, 2, 2, 6, 9, 1, 1, 1, 3, 1, 2, 6, 1, 5, 1, 1, 2, 1, 13, 2, 2, 5, 1, 2, 2, 1, 5, 1, 3, 1, 3, 1, 2, 2, 2, 2, 8, 3, 1, 2, 2, 1, 10, 2, 2, 2, 3, 3, 1, 7, 1, 8, 3, 1, 1, 1, 1, 1, 1, 1, 1, 5, 2, 1, 2, 17, 1, 4, 31, 2, 2, 5, 30, 1, 8, 2, 1

Offset: 0

N. J. A. Sloane, Feb 21 2000

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).

			1.19814023473559220743992249228...

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

Harry J. Smith, Table of n, a(n) for n = 0..19999
Eric Weisstein's World of Mathematics, Gauss's Constant.
G. Xiao, Contfrac
Index entries for continued fractions for constants

Cf. A014549, A053002 without the leading term, A053004 (decimal expansion).

ContinuedFraction[ArithmeticGeometricMean[1,Sqrt[2]],100] (* Harvey P. Dale, Feb 26 2012 *)

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(agm(1, sqrt(2))); for (n=1, 20000, write("b053003.txt", n-1, " ", x[n])); } \\ Harry J. Smith, Apr 20 2009

More terms from James Sellers, Feb 22 2000

Offset changed by Andrew Howroyd, Aug 03 2024

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A014549 Decimal expansion of 1 / M(1,sqrt(2)) (Gauss's constant).

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

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A053003 Continued fraction for M(1,sqrt(2)).

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