A243308 Decimal expansion of h_3, a constant related to certain evaluations of the gamma function from elliptic integrals.

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

Offset

1,4

References

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 1.5.4 Gamma function, p. 34.

A. P. Prudnikov and Yu. A. Brychkov and O. I. Marichev, Integrals and Series vol 3 (1990) eq. 7.3.1.41

Links

Table of n, a(n) for n=1..103.

Eric Weisstein's MathWorld, Gamma function

Formula

4*K(4*sqrt(3)-7)/(sqrt(2+sqrt(3))*Pi), where K is the complete elliptic integral of the first kind.

3^(1/4)*GAMMA(1/3)^3/(2*2^(1/3)*Pi^2), where GAMMA is the Euler Gamma function.

GAMMA(1/6)^(3/2)/(2^(5/6)*sqrt(3)*Pi^(5/4)).

Equals 2F1(1/4,1/4 ; 1 ; 1/4). - R. J. Mathar, Nov 03 2025

Example

1.0174087975959560086695387533500634259952569...

Maple

Re(evalf(4*EllipticK(sqrt((4*sqrt(3)-7)))/(sqrt(2+sqrt(3))*Pi), 120)); # Vaclav Kotesovec, Apr 22 2015
LegendreP(-1/2, 0, sqrt(3/4)) ; evalf(%) ; # R. J. Mathar, Nov 03 2025

Mathematica

RealDigits[4*EllipticK[4*Sqrt[3]-7]/(Sqrt[2+Sqrt[3]]*Pi), 10, 103] // First
RealDigits[1/ArithmeticGeometricMean[1, Sqrt[2 + Sqrt[3]]/2], 10, 103][[1]] (* Jan Mangaldan, Jan 06 2017 *)
RealDigits[2 EllipticK[(2 - Sqrt[3])/4]/Pi, 10, 103][[1]] (* Jan Mangaldan, Jan 06 2017 *)

Crossrefs

Cf. A073005, A175379, A175574, A263490 (squared).

Sequence in context: A196624 A157413 A258500 * A293609 A294514 A099935

Adjacent sequences: A243305 A243306 A243307 * A243309 A243310 A243311

Keyword

nonn,cons,easy

Author

Jean-François Alcover, Jun 03 2014

Status

approved