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A256514 Decimal expansion of the amplitude of a simple pendulum the period of which is twice the period in the small-amplitude approximation. 0
2, 7, 8, 8, 2, 3, 1, 1, 2, 4, 1, 0, 7, 2, 0, 4, 3, 0, 1, 4, 2, 1, 5, 2, 1, 8, 4, 7, 5, 3, 0, 8, 9, 0, 7, 2, 7, 6, 1, 5, 9, 0, 8, 7, 2, 5, 4, 6, 4, 9, 4, 9, 3, 0, 5, 4, 6, 8, 7, 1, 8, 8, 5, 6, 6, 6, 0, 6, 7, 2, 2, 6, 5, 6, 5, 9, 0, 5, 8, 0, 4, 4, 7, 2, 5, 0, 2, 7, 9, 1, 7, 5, 7, 8, 8, 4, 0, 6, 7, 5, 7, 2 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,1

LINKS

Table of n, a(n) for n=1..102.

Claudio G. Carvalhaes, Patrick Suppes, Approximations for the period of the simple pendulum based on the arithmetic-geometric mean

Eric Weisstein's MathWorld, Arithmetic-Geometric Mean

Eric Weisstein's MathWorld, Complete Elliptic Integral of the First Kind

Wikipedia, Pendulum

FORMULA

Solution to (2*K(sin(a/2)^2))/Pi = 2, where K is the complete elliptic integral of the first kind.

Also solution to 1/AGM(1, cos(a/2)) = 2, where AGM is the arithmetic-geometric mean.

EXAMPLE

2.7882311241072043014215218475308907276159087254649493...

= 159.75387571836004625994511811959034206912586138415864587... in degrees.

MATHEMATICA

a2 = a /. FindRoot[ (2*EllipticK[ Sin[a/2]^2 ])/Pi == 2, {a, 3}, WorkingPrecision -> 102]; RealDigits[a2] // First

PROG

(PARI) solve(x=2, 3, 1/agm(cos(x/2), 1)-2) \\ Charles R Greathouse IV, Mar 03 2016

CROSSREFS

Cf. A175574, A226204.

Sequence in context: A102268 A155062 A021786 * A155982 A152778 A043054

Adjacent sequences:  A256511 A256512 A256513 * A256515 A256516 A256517

KEYWORD

nonn,cons,easy

AUTHOR

Jean-Fran├žois Alcover, Apr 01 2015

STATUS

approved

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Last modified October 21 11:42 EDT 2019. Contains 328296 sequences. (Running on oeis4.)