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

Author

Jean-François Alcover, Apr 01 2015

Keywords

• nonn
• cons
• easy

Examples

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

Links

• 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.

Crossrefs

Cf. A175574, A226204.

Programs

Mathematica

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

PARI

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

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.