A309893 -id:A309893 - cp's OEIS frontend

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

Daniel Hoyt, Aug 26 2019

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.

			0.9019793381143431233972715365...

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

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

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

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))

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

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

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

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