A290570 Decimal expansion of Integral_{0..Pi/2} dtheta/(cos(theta)^3 + sin(theta)^3)^(2/3).

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

Offset: 1

Author

Jean-François Alcover, Aug 07 2017

Keywords

• nonn
• cons

Examples

			1.766638750285449957313689499648438702571868538202557530126905241835453...
		

References

• Oscar S. Adams, Elliptic Functions Applied to Conformal World Maps, Special Publication No. 112 of the U.S. Coast and Geodetic Survey, 1925. See constant K p. 9 and previous pages.

Links

• StackExchange, Question 2380131 Evaluate in closed form Integral_{0..Pi/2} dtheta/(cos(theta)^3+sin(theta)^3)^(2/3), August 2017.

Crossrefs

Cf. A073005 (Gamma(1/3)), A073006 (Gamma(2/3)), A197374 (Beta(1/3,1/3)).

Cf. A104133, A104134, A197374.

Programs

Mathematica

RealDigits[(1/3)*Gamma[1/3]^2/Gamma[2/3], 10, 105]

PARI

(1/3)*gamma(1/3)^2/gamma(2/3) \\ Michel Marcus, Aug 07 2017

Formula

Equals (1/3)*Beta(1/3,1/3).

Equals (1/3)*Gamma(1/3)^2/Gamma(2/3).

Equals A197374/3. - Michel Marcus, Jun 08 2020

From Peter Bala, Mar 01 2022: (Start)

Equals 2*Sum_{n >= 0} (1/(3*n+1) + 1/(3*n-2))*binomial(1/3,n). Cf. A002580 and A175576.

Equals Sum_{n >= 0} (-1)^n*(1/(3*n+1) - 1/(3*n-2))*binomial(1/3,n).

Equals hypergeom([1/3, 2/3], [4/3], 1) = (3/2)*hypergeom([-1/3, -2/3], [4/3], 1) = 2*hypergeom([1/3, 2/3], [4/3], -1) = hypergeom([-1/3, -2/3, 5/6], [4/3, -1/6], -1). (End)