A062539 Decimal expansion of the Lemniscate constant or Gauss's constant.

2, 6, 2, 2, 0, 5, 7, 5, 5, 4, 2, 9, 2, 1, 1, 9, 8, 1, 0, 4, 6, 4, 8, 3, 9, 5, 8, 9, 8, 9, 1, 1, 1, 9, 4, 1, 3, 6, 8, 2, 7, 5, 4, 9, 5, 1, 4, 3, 1, 6, 2, 3, 1, 6, 2, 8, 1, 6, 8, 2, 1, 7, 0, 3, 8, 0, 0, 7, 9, 0, 5, 8, 7, 0, 7, 0, 4, 1, 4, 2, 5, 0, 2, 3, 0, 2, 9, 5, 5, 3, 2, 9, 6, 1, 4, 2, 9, 0, 9, 3, 4, 4, 6, 1, 3

Offset

1,1

References

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Sections 2.3 and 6.2, pp. 99, 420.

Links

Harry J. Smith, Table of n, a(n) for n = 1..5000

Simon Plouffe, Lemniscate or Gauss constant.

Simon Plouffe, Lemniscate constant or Gauss constant.

Eric Weisstein's World of Mathematics, Lemniscate Constant.

Wikipedia, Lemniscate constant.

Index entries for transcendental numbers.

Formula

Equals (1/2)*sqrt(2*Pi^3)/Gamma(3/4)^2.

A093341 multiplied by A002193. - R. J. Mathar, Aug 28 2013

From Martin Renner, Aug 16 2019: (Start)

Equals 2*Integral_{x=0..1} 1/sqrt(1-x^4) dx.

Equals 1/2*B(1/4,1/2) with Beta function B(x,y) = Gamma(x)*Gamma(y)/Gamma(x+y). (End)

Equals Pi/AGM(1, sqrt(2)). - Jean-François Alcover, Feb 28 2021

Equals 2*hypergeom([1/2, 1/4], [5/4], 1). - Peter Bala, Mar 02 2022

Equals (1/2)*A064853 = 2*A085565. - Amiram Eldar, May 04 2022

Equals Pi*A014549. - Hugo Pfoertner, Jun 28 2024

Equals Integral_{x=0..Pi} 1/sqrt(1 + sin(x)^2) dx = EllipticK(-1) (see Finch at p. 420). - Stefano Spezia, Dec 15 2024

Example

2.622057554292119810464839589891119413682754951431623162816821703...

Maple

evalf((1/2)*sqrt(2*Pi^3)/GAMMA(3/4)^2, 120); # Muniru A Asiru, Oct 08 2018
evalf(1/2*GAMMA(1/4)*GAMMA(1/2)/GAMMA(3/4), 120); # Martin Renner, Aug 16 2019
evalf(1/2*Beta(1/4, 1/2), 120); # Martin Renner, Aug 16 2019
evalf(2*int(1/sqrt(1-x^4), x=0..1), 120); # Martin Renner, Aug 16 2019

Mathematica

RealDigits[Pi^(3/2)/Gamma[3/4]^2*2^(1/2)/2, 10, 111][[1]] (* Robert G. Wilson v, May 19 2004 *)

Prog

(PARI) print(1/2*Pi^(3/2)/gamma(3/4)^2*2^(1/2))
(PARI) allocatemem(932245000); default(realprecision, 5080); x=Pi^(3/2)*sqrt(2)/(2*gamma(3/4)^2); for (n=1, 5000, d=floor(x); x=(x-d)*10; write("b062539.txt", n, " ", d)); \\ Harry J. Smith, Jun 20 2009
(PARI) Pi/agm(1, sqrt(2)) \\ Charles R Greathouse IV, Feb 04 2015
(PARI) intnum(x=0, Pi, 1/sqrt(1 + sin(x)^2)) \\ Charles R Greathouse IV, Feb 04 2025
(Magma) SetDefaultRealField(RealField(100)); R:= RealField(); (1/2)*Sqrt(2*Pi(R)^3)/Gamma(3/4)^2; // G. C. Greubel, Oct 07 2018

Crossrefs

Cf. A062540, A064853, A085565.

Cf. A002193, A014549, A093341.

Equals A000796/A053004 (see PARI script).

Sequence in context: A303335 A332390 A289382 * A064136 A347238 A171898

Adjacent sequences: A062536 A062537 A062538 * A062540 A062541 A062542

Keyword

nonn,cons,easy

Author

Jason Earls, Jun 25 2001

Status

approved