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%I #39 Sep 24 2022 08:17:51
%S 5,2,4,4,1,1,5,1,0,8,5,8,4,2,3,9,6,2,0,9,2,9,6,7,9,1,7,9,7,8,2,2,3,8,
%T 8,2,7,3,6,5,5,0,9,9,0,2,8,6,3,2,4,6,3,2,5,6,3,3,6,4,3,4,0,7,6,0,1,5,
%U 8,1,1,7,4,1,4,0,8,2,8,5,0,0,4,6,0,5,9,1,0,6,5,9,2,2,8,5,8,1,8,6,8,9
%N Lemniscate constant.
%H Harry J. Smith, <a href="/A064853/b064853.txt">Table of n, a(n) for n = 1..5000</a>
%H Markus Faulhuber, Anupam Gumber, and Irina Shafkulovska, <a href="https://arxiv.org/abs/2209.04202">The AGM of Gauss, Ramanujan's corresponding theory, and spectral bounds of self-adjoint operators</a>, arXiv:2209.04202 [math.CA], 2022, p. 15.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LemniscateConstant.html">Lemniscate Constant</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Lemniscate.html">Lemniscate</a>.
%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%F Equals Gamma(1/4)^2/sqrt(2*Pi). - _G. C. Greubel_, Oct 07 2018
%F Equals 2*A062539 = 4*A085565. - _Amiram Eldar_, May 04 2022
%F From _Stefano Spezia_, Sep 23 2022: (Start)
%F Equals 4*Integral_{x=0..Pi/2} 1/sqrt(2*(1 - (1/2)*sin(x)^2)) dx [Gauss, 1799] (see Faulhuber et al.).
%F Equals 2*sqrt(2)*A093341. (End)
%e 5.244115108584239620929679...
%t First@RealDigits[ N[ Gamma[ 1/4 ]^2/Sqrt[ 2 Pi ], 102 ] ]
%o (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
%o (PARI) gamma(1/2)*gamma(1/4)/gamma(3/4) \\ _Charles R Greathouse IV_, Oct 29 2021
%o (Magma) SetDefaultRealField(RealField(100)); R:= RealField(); Gamma(1/4)^2/Sqrt(2*Pi(R)); // _G. C. Greubel_, Oct 07 2018
%Y Cf. A002193, A019727, A062539, A068466, A085565, A093341.
%K nonn,cons,easy
%O 1,1
%A _Eric W. Weisstein_, Sep 22 2001
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