cp's OEIS Frontend

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

%I A096427 #82 Feb 16 2025 08:32:53
%S A096427 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,
%T A096427 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,
%U A096427 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
%N A096427 Decimal expansion of 1/(sqrt(2)*G), where G is Gauss's constant A014549.
%C A096427 Also, decimal expansion of Product_{n>=1} (1-1/(4n-1)^2). - _Bruno Berselli_, Apr 02 2013
%D A096427 Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 6.1, p. 421.
%D A096427 Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 1, equation 1:7:6 at page 13.
%H A096427 Harry J. Smith, <a href="/A096427/b096427.txt">Table of n, a(n) for n = 0..20000</a>
%H A096427 Peter Bala, <a href="/A096427/a096427.pdf">Notes on the constants A096427 and A224268 </a>.
%H A096427 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GausssConstant.html">Gauss's Constant</a>.
%H A096427 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>.
%H A096427 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/UbiquitousConstant.html">Ubiquitous Constant</a>.
%H A096427 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%F A096427 Also equals agm(1,1/sqrt(2)) since agm(1,1/b) = (1/b)*agm(1,b). - _Gerald McGarvey_, Sep 22 2008
%F A096427 From _Peter Bala_, Feb 26 2019: (Start)
%F A096427 C = Gamma(3/4)^2/sqrt(Pi).
%F A096427 C = 1/( Sum_{n = -inf..inf} exp(-Pi*n^2) )^2.
%F A096427 C = (1/sqrt(2)) * 1/( Sum_{n = -inf..inf} (-1)^n*exp(-Pi*n^2 ) )^2.
%F A096427 Conjecturally, C = (1/sqrt(2)) * 1/( Sum_{n = -inf..inf} exp(-Pi*(n+1/2)^2 ) )^2.
%F A096427 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,....
%F A096427 C = 1 - Integral_{x = 0..1} (sqrt(1 + x^4) - 1)/x^2 dx.
%F A096427 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).
%F A096427 Continued fraction: 1 - 1/(3 + 6/(1 + 12/(3 + ... + (4*n - 1)*(4*n - 2)/(1 + 4*n*(4*n - 1)/(3 + ... ))))). (End)
%F A096427 From _Peter Bala_, Mar 02 2022 : (Start)
%F A096427 C = (2/3)*hypergeom([1/4, 3/4], [7/4], 1)
%F A096427 C = hypergeom([-1/4, 1/4], [3/4], 1).
%F A096427 C = hypergeom([-1/2, -1/4], [3/4], -1). Cf. A053004.
%F A096427 C = (16/21)*hypergeom([-1/4, -3/4], [7/4], 1). (End)
%F A096427 Equals Pi/(sqrt(2)*A062539). - _Amiram Eldar_, May 04 2022
%F A096427 C = Integral_{x = 0..Pi/2} sqrt(sin(x)*cos(x)) dx. - _Adam Hugill_, Nov 27 2022
%F A096427 Equals 1/A175574 = sqrt(A293238) = A327995^2. - _Hugo Pfoertner_, Dec 26 2024
%e A096427 0.8472130847939790866064991234821916364814459103269... = agm(1, sqrt(1/2))
%t A096427 RealDigits[ArithmeticGeometricMean[1, Sqrt[2]]/Sqrt[2], 10, 110][[1]] (* _Bruno Berselli_, Apr 02 2013 *)
%t A096427 (* From the comment: *) RealDigits[N[Product[1 - 1/(4 n - 1)^2, {n, 1, Infinity}], 110]][[1]] (* _Bruno Berselli_, Apr 02 2013 *)
%o A096427 (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
%o A096427 (PARI) agm(1, sqrt(1/2)) \\ _Michel Marcus_, Jun 09 2019
%o A096427 (Magma) SetDefaultRealField(RealField(100)); R:= RealField(); Gamma(3/4)^2/(Sqrt(2)*Sqrt(Pi(R)/2)); // _G. C. Greubel_, Aug 17 2018
%Y A096427 Cf. A014549, A062539, A224268, A091670 (1/C^2), A175574 (1/C), A293238 (C^2), A053004 (sqrt(2)*C), A327995.
%K A096427 nonn,cons,easy
%O A096427 0,1
%A A096427 _Eric W. Weisstein_, Jul 21 2004