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A256514
Decimal expansion of the amplitude of a simple pendulum the period of which is twice the period in the small-amplitude approximation.
1
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
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
Sequence in context: A102268 A155062 A021786 * A372340 A369988 A155982
KEYWORD
nonn,cons,easy
AUTHOR
STATUS
approved