A275486 Decimal expansion of Pi_3, the analog of Pi for generalized trigonometric functions of order p=3.

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

Offset: 1

Author

Jean-François Alcover, Jul 30 2016

Keywords

• nonn
• cons
• easy

Comments

The area of the circumcircle of a unit-area equilateral triangle. - Amiram Eldar, Aug 13 2020

Examples

			2.41839915231229046745877101018954097637875499745698743409317991385...
		

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.

Crossrefs

Cf. A240935 (reciprocal), A263498.

Programs

Mathematica

RealDigits[4 Pi/(3 Sqrt[3]), 10, 102][[1]]

PARI

4*Pi/sqrt(27) \\ Charles R Greathouse IV, Aug 01 2016

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