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%I #11 Jun 29 2023 09:45:52
%S 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,
%T 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,
%U 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
%N Decimal expansion of the amplitude of a simple pendulum the period of which is twice the period in the small-amplitude approximation.
%H Claudio Carvalhaes and Patrick Suppes, <a href="https://doi.org/10.1119/1.2968864">Approximations for the period of the simple pendulum based on the arithmetic-geometric mean</a>, American Journal of Physics 76, 1150-1154 (2008).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Arithmetic-GeometricMean.html">Arithmetic-Geometric Mean</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CompleteEllipticIntegraloftheFirstKind.html">Complete Elliptic Integral of the First Kind</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Pendulum_(mathematics)">Pendulum</a>.
%F Solution to (2*K(sin(a/2)^2))/Pi = 2, where K is the complete elliptic integral of the first kind.
%F Also solution to 1/AGM(1, cos(a/2)) = 2, where AGM is the arithmetic-geometric mean.
%e 2.7882311241072043014215218475308907276159087254649493...
%e = 159.75387571836004625994511811959034206912586138415864587... in degrees.
%t a2 = a /. FindRoot[ (2*EllipticK[ Sin[a/2]^2 ])/Pi == 2, {a, 3}, WorkingPrecision -> 102]; RealDigits[a2] // First
%o (PARI) solve(x=2,3,1/agm(cos(x/2),1)-2) \\ _Charles R Greathouse IV_, Mar 03 2016
%Y Cf. A175574, A226204.
%K nonn,cons,easy
%O 1,1
%A _Jean-François Alcover_, Apr 01 2015
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