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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 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.)