A014549 - OEIS

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

(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.

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.5.4, p. 34.

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.

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

EXAMPLE

0.8346268416740731862814297327990468...

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 *)

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

(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

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

CROSSREFS

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

Sequence in context: A222232 A091895 A111436 * A346441 A334064 A021549

Adjacent sequences: A014546 A014547 A014548 * A014550 A014551 A014552

KEYWORD