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

Author

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

Keywords

• nonn
• cons
• nice

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).

Examples

			0.8346268416740731862814297327990468...
		

References

• 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.

Links

• 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.

Crossrefs

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

Programs

Magma

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

Maple

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

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]] (* 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 *)

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

Python

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

Formula

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

Extensions

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

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