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Decimal expansion of Integral_{0..Pi/2} dtheta/(cos(theta)^3 + sin(theta)^3)^(2/3).
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%I #39 Mar 03 2022 11:45:04

%S 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,

%T 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,

%U 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

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

%D 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.

%H StackExchange, <a href="https://math.stackexchange.com/questions/2380131/evaluate-int-0-pi-2-fracd-theta-left-cos3-theta-sin3-theta-r">Question 2380131</a> Evaluate in closed form Integral_{0..Pi/2} dtheta/(cos(theta)^3+sin(theta)^3)^(2/3), August 2017.

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

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

%F Equals A197374/3. - _Michel Marcus_, Jun 08 2020

%F From _Peter Bala_, Mar 01 2022: (Start)

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

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

%F 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)

%e 1.766638750285449957313689499648438702571868538202557530126905241835453...

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

%o (PARI) (1/3)*gamma(1/3)^2/gamma(2/3) \\ _Michel Marcus_, Aug 07 2017

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

%Y Cf. A104133, A104134, A197374.

%K nonn,cons

%O 1,2

%A _Jean-François Alcover_, Aug 07 2017