A309893 Decimal expansion of AGM(1, sqrt(3)/2).

9, 3, 1, 8, 0, 8, 3, 9, 1, 6, 2, 2, 4, 4, 8, 2, 7, 1, 1, 7, 7, 8, 4, 4, 5, 1, 5, 5, 1, 2, 1, 3, 5, 2, 9, 7, 5, 7, 8, 7, 6, 6, 4, 2, 8, 4, 1, 3, 4, 2, 6, 8, 6, 1, 1, 1, 0, 2, 2, 0, 6, 1, 3, 4, 8, 9, 1, 6, 2, 8, 9, 7, 2, 8

Offset

0,1

Comments

Related to the pendulum acceleration relation at 60 degrees. In general, the period T of a mathematical pendulum with a maximum deflection angle theta is 2*Pi*sqrt(L/g)/AGM(1, cos(theta/2)), where L is the length of the pendulum, g is the gravitational acceleration, and 0 < theta <= 90 degrees. For theta = 60 degrees, the period is T = 2*Pi*sqrt(L/g)/AGM(1, sqrt(3)/2). - Jianing Song, Nov 21 2022

Links

Table of n, a(n) for n=0..73.

Wikipedia, Pendulum (mechanics)

Formula

AGM(1, sin(Pi/3)).

Example

0.931808391622448271177844...

Mathematica

RealDigits[ArithmeticGeometricMean[1, Sqrt[3]/2], 10, 100][[1]] (* Amiram Eldar, Aug 21 2019 *)

Prog

(Python)
import decimal
prec = int(input('Precision: '))
decimal.getcontext().prec = prec
D = decimal.Decimal
def agm(a, b):
    for x in range(prec):
        a, b = (a + b) / 2, (a * b).sqrt()
    return a
print(agm(1, D(3).sqrt()/2))
(PARI) agm(1, sqrt(3)/2) \\ Michel Marcus, Aug 22 2019
(Sage)
RealField(300)(1.0).agm(sqrt(3)/2) # Peter Luschny, Aug 22 2019

Crossrefs

Cf. A310000 (AGM(1, cos(Pi/5))), A096427 (AGM(1, sqrt(2)/2)), A053004, A014549, A068521.

Sequence in context: A081813 A197003 A048799 * A188887 A250091 A199152

Adjacent sequences: A309890 A309891 A309892 * A309894 A309895 A309896

Keyword

nonn,cons

Author

Daniel Hoyt, Aug 21 2019

Status

approved