A096427 Decimal expansion of 1/(sqrt(2)*G), where G is Gauss's constant A014549.

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

Offset: 0

Author

Eric W. Weisstein, Jul 21 2004

Keywords

• nonn
• cons
• easy

Comments

Also, decimal expansion of Product_{n>=1} (1-1/(4n-1)^2). - Bruno Berselli, Apr 02 2013

Examples

			0.8472130847939790866064991234821916364814459103269... = agm(1, sqrt(1/2))
		

References

• Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 6.1, p. 421.
• Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 1, equation 1:7:6 at page 13.

Links

• Harry J. Smith, Table of n, a(n) for n = 0..20000
• Peter Bala, Notes on the constants A096427 and A224268 .
• Eric Weisstein's World of Mathematics, Gauss's Constant.
• Eric Weisstein's World of Mathematics, Jacobi Theta Functions.
• Eric Weisstein's World of Mathematics, Ubiquitous Constant.
• Index entries for transcendental numbers.

Crossrefs

Cf. A014549, A062539, A224268, A091670 (1/C^2), A175574 (1/C), A293238 (C^2), A053004 (sqrt(2)*C), A327995.

Programs

Magma

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

Mathematica

RealDigits[ArithmeticGeometricMean[1, Sqrt[2]]/Sqrt[2], 10, 110][[1]] (* Bruno Berselli, Apr 02 2013 *)
(* From the comment: *) RealDigits[N[Product[1 - 1/(4 n - 1)^2, {n, 1, Infinity}], 110]][[1]] (* Bruno Berselli, Apr 02 2013 *)

PARI

{ default(realprecision, 20080); x=agm(1, sqrt(1/2)); d=0; for (n=0, 20000, x=(x-d)*10; d=floor(x); write("b096427.txt", n, " ", d)); } \\ Harry J. Smith, Apr 15 2009

PARI

agm(1, sqrt(1/2)) \\ Michel Marcus, Jun 09 2019

Formula

Also equals agm(1,1/sqrt(2)) since agm(1,1/b) = (1/b)*agm(1,b). - Gerald McGarvey, Sep 22 2008

From Peter Bala, Feb 26 2019: (Start)

C = Gamma(3/4)^2/sqrt(Pi).

C = 1/( Sum_{n = -inf..inf} exp(-Pi*n^2) )^2.

C = (1/sqrt(2)) * 1/( Sum_{n = -inf..inf} (-1)^n*exp(-Pi*n^2 ) )^2.

Conjecturally, C = (1/sqrt(2)) * 1/( Sum_{n = -inf..inf} exp(-Pi*(n+1/2)^2 ) )^2.

C = ((-1)^m*4^m/binomial(2*m,m)) * Product_{n >= 0} ( 1 - (4*m + 1)^2/(4*n + 3)^2 ), for m = 0,1,2,....

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

C = 1 - Sum_{n >= 1} binomial(1/2,n)/(4*n - 1) = 1 - Sum_{n >= 0} (-1)^n/(4*n + 3)*Catalan(n)/2^(2*n + 1).

Continued fraction: 1 - 1/(3 + 6/(1 + 12/(3 + ... + (4*n - 1)*(4*n - 2)/(1 + 4*n*(4*n - 1)/(3 + ... ))))). (End)

From Peter Bala, Mar 02 2022 : (Start)

C = (2/3)*hypergeom([1/4, 3/4], [7/4], 1)

C = hypergeom([-1/4, 1/4], [3/4], 1).

C = hypergeom([-1/2, -1/4], [3/4], -1). Cf. A053004.

C = (16/21)*hypergeom([-1/4, -3/4], [7/4], 1). (End)

Equals Pi/(sqrt(2)*A062539). - Amiram Eldar, May 04 2022

C = Integral_{x = 0..Pi/2} sqrt(sin(x)*cos(x)) dx. - Adam Hugill, Nov 27 2022

Equals 1/A175574 = sqrt(A293238) = A327995^2. - Hugo Pfoertner, Dec 26 2024