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A310000 Decimal expansion of AGM(1, phi/2), where phi is the golden ratio (A001622). 0
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; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

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.

LINKS

Table of n, a(n) for n=0..84.

FORMULA

Equals AGM(1, cos(Pi/5)).

EXAMPLE

0.9019793381143431233972715365...

MATHEMATICA

RealDigits[ArithmeticGeometricMean[1, GoldenRatio/2], 10, 100][[1]] (* Amiram Eldar, Aug 26 2019 *)

PROG

(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))

CROSSREFS

Cf. A001622, A309893, A053004, A014549, A068521.

Sequence in context: A249418 A256036 A065471 * A199284 A189186 A221429

Adjacent sequences:  A309997 A309998 A309999 * A310001 A310002 A310003

KEYWORD

nonn,cons

AUTHOR

Daniel Hoyt, Aug 26 2019

STATUS

approved

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Last modified February 24 22:57 EST 2020. Contains 332216 sequences. (Running on oeis4.)