OFFSET
1,1
LINKS
Harry J. Smith, Table of n, a(n) for n = 1..5000
Markus Faulhuber, Anupam Gumber, and Irina Shafkulovska, The AGM of Gauss, Ramanujan's corresponding theory, and spectral bounds of self-adjoint operators, arXiv:2209.04202 [math.CA], 2022, p. 15.
Eric Weisstein's World of Mathematics, Lemniscate Constant.
Eric Weisstein's World of Mathematics, Lemniscate.
FORMULA
Equals Gamma(1/4)^2/sqrt(2*Pi). - G. C. Greubel, Oct 07 2018
From Stefano Spezia, Sep 23 2022: (Start)
Equals 4*Integral_{x=0..Pi/2} 1/sqrt(2*(1 - (1/2)*sin(x)^2)) dx [Gauss, 1799] (see Faulhuber et al.).
Equals 2*sqrt(2)*A093341. (End)
EXAMPLE
5.244115108584239620929679...
MATHEMATICA
First@RealDigits[ N[ Gamma[ 1/4 ]^2/Sqrt[ 2 Pi ], 102 ] ]
PROG
(PARI) { allocatemem(932245000); default(realprecision, 5080); x=gamma(1/4)^2/sqrt(2*Pi); for (n=1, 5000, d=floor(x); x=(x-d)*10; write("b064853.txt", n, " ", d)); } \\ Harry J. Smith, Jun 20 2009
(PARI) gamma(1/2)*gamma(1/4)/gamma(3/4) \\ Charles R Greathouse IV, Oct 29 2021
(Magma) SetDefaultRealField(RealField(100)); R:= RealField(); Gamma(1/4)^2/Sqrt(2*Pi(R)); // G. C. Greubel, Oct 07 2018
CROSSREFS
KEYWORD
AUTHOR
Eric W. Weisstein, Sep 22 2001
STATUS
approved