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A310000
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Decimal expansion of AGM(1, phi/2), where phi is the golden ratio (A001622).
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1
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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, 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, 1, 4, 2, 6, 2, 6, 2, 0, 5, 7, 7, 2, 7, 5, 4, 1, 8
(list;
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OFFSET
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0,1
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COMMENTS
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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.
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LINKS
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Table of n, a(n) for n=0..84.
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FORMULA
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Equals AGM(1, cos(Pi/5)).
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EXAMPLE
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0.9019793381143431233972715365...
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MATHEMATICA
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RealDigits[ArithmeticGeometricMean[1, GoldenRatio/2], 10, 100][[1]] (* Amiram Eldar, Aug 26 2019 *)
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PROG
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(Python3)
import decimal
iters = int(input('Precision: '))
decimal.getcontext().prec = iters
D = decimal.Decimal
def agm(a, b):
for x in range(iters):
a, b = (a + b) / 2, (a * b).sqrt()
return a
print(agm(1, (D(5).sqrt()+1)/4))
(PARI) agm(1, cos(Pi/5)) \\ Michel Marcus, Apr 05 2020
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CROSSREFS
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Cf. A001622, A309893, A053004, A014549, A068521.
Sequence in context: A249418 A256036 A065471 * A199284 A189186 A221429
Adjacent sequences: A309997 A309998 A309999 * A310001 A310002 A310003
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KEYWORD
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nonn,cons
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AUTHOR
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Daniel Hoyt, Aug 26 2019
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STATUS
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approved
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