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

0,1

Comments

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

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.

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

Example

0.8472130847939790866064991234821916364814459103269... = agm(1, sqrt(1/2)) - Harry J. Smith, Apr 15 2009

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 *)

Prog

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

Crossrefs

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

Sequence in context: A322743 A168546 A195346 * A176453 A257775 A242023

Adjacent sequences: A096424 A096425 A096426 * A096428 A096429 A096430

Keyword

nonn,cons,easy

Author

Eric W. Weisstein, Jul 21 2004

Status

approved