A064853 Lemniscate constant.

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, 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, 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

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.

Index entries for transcendental numbers.

Formula

Equals Gamma(1/4)^2/sqrt(2*Pi). - G. C. Greubel, Oct 07 2018

Equals 2*A062539 = 4*A085565. - Amiram Eldar, May 04 2022

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

Cf. A002193, A019727, A062539, A068466, A085565, A093341.

Sequence in context: A249920 A021660 A266122 * A177148 A188739 A308171

Adjacent sequences: A064850 A064851 A064852 * A064854 A064855 A064856

Keyword

nonn,cons,easy

Author

Eric W. Weisstein, Sep 22 2001

Status

approved