A274014 - OEIS

%I #22 Aug 26 2021 03:24:11

%S 6,1,7,6,6,0,1,9,8,7,6,5,8,6,9,3,4,6,4,7,4,5,6,8,4,0,8,4,1,0,7,3,7,4,

%T 4,1,7,5,7,5,3,7,2,3,4,3,4,6,9,6,1,2,5,1,0,2,9,1,1,4,4,1,9,2,2,5,4,1,

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

%N Decimal expansion of the arc length of an ellipse with semi-major axis 1 and eccentricity sin(Pi/12), an arc length which evaluates without using elliptic integrals (a computation due to Ramanujan).

%D Richard E. Crandall, Projects in Scientific Computation, Springer, 1994; see p. 48.

%H G. C. Greubel, <a href="/A274014/b274014.txt">Table of n, a(n) for n = 1..5000</a>

%F Equals (2*((6 + sqrt(3) + 4*sqrt(2 + sqrt(3)))*E((-2 + sqrt(2 + sqrt(3)))^2/(2 + sqrt(2 + sqrt(3)))^2) - 4*sqrt(2 + sqrt(3))*K((-2 + sqrt(2 + sqrt(3)))^2/ (2 + sqrt(2 + sqrt(3)))^2)))/(2 + sqrt(2 + sqrt(3))), where K and E are the elliptic integrals of first and second kind.

%F Equals sqrt(Pi/sqrt(3))*(((1 + 1/sqrt(3))*Gamma(1/3))/Gamma(5/6) + (2*Gamma(5/6))/Gamma(1/3)).

%e 6.176601987658693464745684084107374417575372343469612510291144192254...

%t p = Sqrt[Pi/Sqrt[3]]*((1 + 1/Sqrt[3])*Gamma[1/3]/Gamma[5/6] + 2*Gamma[5/6]/ Gamma[1/3]);

%t RealDigits[p, 10, 103][[1]]

%o (PARI) sqrt(Pi/sqrt(3))*((1 + 1/sqrt(3))*gamma(1/3)/gamma(5/6) + 2*gamma(5/6)/gamma(1/3)) \\ _G. C. Greubel, Jun 05 2017

%Y Cf. A019824.

%K nonn,cons

%O 1,1

%A _Jean-François Alcover_, Jun 10 2016