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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).
2
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
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
KEYWORD
nonn,cons
AUTHOR
STATUS
approved