A256514 Decimal expansion of the amplitude of a simple pendulum the period of which is twice the period in the small-amplitude approximation.

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

Offset

1,1

Links

Table of n, a(n) for n=1..102.

Claudio Carvalhaes and Patrick Suppes, Approximations for the period of the simple pendulum based on the arithmetic-geometric mean, American Journal of Physics 76, 1150-1154 (2008).

Eric Weisstein's World of Mathematics, Arithmetic-Geometric Mean.

Eric Weisstein's World of Mathematics, Complete Elliptic Integral of the First Kind.

Wikipedia, Pendulum.

Formula

Solution to (2*K(sin(a/2)^2))/Pi = 2, where K is the complete elliptic integral of the first kind.

Also solution to 1/AGM(1, cos(a/2)) = 2, where AGM is the arithmetic-geometric mean.

Example

2.7882311241072043014215218475308907276159087254649493...
= 159.75387571836004625994511811959034206912586138415864587... in degrees.

Mathematica

a2 = a /. FindRoot[ (2*EllipticK[ Sin[a/2]^2 ])/Pi == 2, {a, 3}, WorkingPrecision -> 102]; RealDigits[a2] // First

Prog

(PARI) solve(x=2, 3, 1/agm(cos(x/2), 1)-2) \\ Charles R Greathouse IV, Mar 03 2016

Crossrefs

Cf. A175574, A226204.

Sequence in context: A102268 A155062 A021786 * A369988 A155982 A152778

Adjacent sequences: A256511 A256512 A256513 * A256515 A256516 A256517

Keyword

nonn,cons,easy

Author

Jean-François Alcover, Apr 01 2015

Status

approved