A310000 -id:A310000 - cp's OEIS frontend

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

Daniel Hoyt, Aug 21 2019

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

			0.931808391622448271177844...

Wikipedia, Pendulum (mechanics)

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

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

agm(1, sqrt(3)/2) \\ Michel Marcus, Aug 22 2019

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))

RealField(300)(1.0).agm(sqrt(3)/2) # Peter Luschny, Aug 22 2019

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

A376643 Decimal expansion 4*EllipticK(4/5)/sqrt(5), where EllipticK is the complete elliptic integral of the first kind.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Oct 01 2024

A point mass is attached to a frictionless pivot by a massless string of length L and revolves in a vertical circle about the pivot in a uniform gravitational field with an acceleration g. The slowest possible motion occurs when the tension in the string is momentarily zero at the top of the route, and the longest-possible period is then c * sqrt(L/g), where c is this constant.

			4.03781163995684643116802887999786493013608399340880...

Physics StackExchange, Maximum period of a vertically spinning ball, 2016.
Physics StackExchange, Maximal time for an object vertically swinging on a rope to complete a full circle, 2023.
Eric Weisstein's World of Mathematics, Complete Elliptic Integral of the First Kind.
Wikipedia, Elliptic integral: Complete elliptic integral of the first kind.

Constants related to similar physical problems: A019692, A038533, A038534, A175574, A256514, A309893, A310000.

RealDigits[4 * EllipticK[4/5] / Sqrt[5], 10, 120][[1]]

PARI
```
4*ellK(sqrt(4/5))/sqrt(5)
```

Equals 2 * Integral_{0..Pi} (1/sqrt(3 + 2*cos(x))) dx.

cp's OEIS Frontend

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

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