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

Author

Daniel Hoyt, Aug 26 2019

Keywords

• nonn
• cons

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.

Examples

			0.9019793381143431233972715365...
		

Crossrefs

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

Programs

Mathematica

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

PARI

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

Python

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))

Formula

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