A310000 Decimal expansion of AGM(1, phi/2), where phi is the golden ratio (A001622).

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

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))
(PARI) agm(1, cos(Pi/5)) \\ Michel Marcus, Apr 05 2020

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