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A096427 Decimal expansion of 1/(sqrt(2)*G), where G is Gauss's constant A014549. 5
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 (list; constant; graph; refs; listen; history; text; internal format)
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

P. Bala, Notes on the constants A096427 and A224268

Eric Weisstein's World of Mathematics, Gauss's Constant

Eric Weisstein's MathWorld, 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)

EXAMPLE

0.847213084793979086606499123482191636481445910327... = 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, 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

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Last modified August 18 22:08 EDT 2019. Contains 326109 sequences. (Running on oeis4.)