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

1,1

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

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)).

Example

6.176601987658693464745684084107374417575372343469612510291144192254...

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]]

Prog

(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

Crossrefs

Cf. A019824.

Sequence in context: A198420 A377343 A110942 * A334962 A082830 A261622

Adjacent sequences: A274011 A274012 A274013 * A274015 A274016 A274017

Keyword

nonn,cons

Author

Jean-François Alcover, Jun 10 2016

Status

approved