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%I #24 Apr 05 2020 13:25:57
%S 9,0,1,9,7,9,3,3,8,1,1,4,3,4,3,1,2,3,3,9,7,2,7,1,5,3,6,5,8,7,7,9,8,6,
%T 2,7,5,5,1,6,2,3,7,4,6,7,3,6,9,9,0,1,4,0,7,9,8,4,7,7,9,4,2,9,1,1,9,4,
%U 1,4,2,6,2,6,2,0,5,7,7,2,7,5,4,1,8
%N Decimal expansion of AGM(1, phi/2), where phi is the golden ratio (A001622).
%C Related to the pendulum acceleration relation at 72 degrees. 2*Pi*sqrt(l/g)/AGM(1, phi/2) gives the period T of a mathematical pendulum with a maximum deflection angle of 72 degrees from the downward vertical. The length of the pendulum is l and g is the gravitational acceleration.
%F Equals AGM(1, cos(Pi/5)).
%e 0.9019793381143431233972715365...
%t RealDigits[ArithmeticGeometricMean[1, GoldenRatio/2], 10, 100][[1]] (* _Amiram Eldar_, Aug 26 2019 *)
%o (Python3)
%o import decimal
%o iters = int(input('Precision: '))
%o decimal.getcontext().prec = iters
%o D = decimal.Decimal
%o def agm(a, b):
%o for x in range(iters):
%o a, b = (a + b) / 2,(a * b).sqrt()
%o return a
%o print(agm(1, (D(5).sqrt()+1)/4))
%o (PARI) agm(1, cos(Pi/5)) \\ _Michel Marcus_, Apr 05 2020
%Y Cf. A001622, A309893, A053004, A014549, A068521.
%K nonn,cons
%O 0,1
%A _Daniel Hoyt_, Aug 26 2019
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