A274014 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).

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, 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, 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

Offset: 1

Author

Jean-François Alcover, Jun 10 2016

Keywords

• nonn
• cons

Examples

			6.176601987658693464745684084107374417575372343469612510291144192254...
		

References

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

Links

• G. C. Greubel, Table of n, a(n) for n = 1..5000

Crossrefs

Cf. A019824.

Programs

Mathematica

p = Sqrt[Pi/Sqrt[3]]*((1 + 1/Sqrt[3])*Gamma[1/3]/Gamma[5/6] + 2*Gamma[5/6]/ Gamma[1/3]);
RealDigits[p, 10, 103][[1]]

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

Formula

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.

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