OFFSET
1,1
COMMENTS
The area of the circumcircle of a unit-area equilateral triangle. - Amiram Eldar, Aug 13 2020
LINKS
David Edmunds and Jan Lang, Generalizing trigonometric functions from different points of view, 2009.
Shingo Takeuchi, A new form of the generalized complete elliptic integrals, arXiv:1411.4778 [math.CA], 2014.
FORMULA
Pi_3 = 2*Pi/(3*sin(Pi/3)) = 2/3 * gamma(1/3) * gamma(2/3) = 4*Pi/(3 * sqrt(3)).
Pi_3 = Sum_{n>=1} 4/(9*n^2 - 9*n + 2).
Pi_3 = 2*Integral_{0..1} (1-x^3)^(-1/3) dx.
Equals 1 + A263498.
Equals Integral_{x=0..oo} 1/(1 + x^(3/2)) dx. - Amiram Eldar, Aug 13 2020
Equals Product_{p prime} (1 + Kronecker(-3, p)/p)^(-1) = Product_{p prime != 3} (1 - (-1)^(p mod 3)/p)^(-1). - Amiram Eldar, Nov 06 2023
EXAMPLE
2.41839915231229046745877101018954097637875499745698743409317991385...
MATHEMATICA
RealDigits[4 Pi/(3 Sqrt[3]), 10, 102][[1]]
PROG
(PARI) 4*Pi/sqrt(27) \\ Charles R Greathouse IV, Aug 01 2016
CROSSREFS
KEYWORD
AUTHOR
Jean-François Alcover, Jul 30 2016
STATUS
approved