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A062539
Decimal expansion of the Lemniscate constant or Gauss's constant.
33
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.
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
Equals A000796/A053004 (see PARI script).
Sequence in context: A303335 A332390 A289382 * A064136 A347238 A171898
KEYWORD
nonn,cons,easy
AUTHOR
Jason Earls, Jun 25 2001
STATUS
approved