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A223067 A sequence related to the period T of a simple gravity pendulum for arbitrary amplitudes. 5
1, 1, 11, 173, 22931, 1319183, 233526463, 2673857519, 39959591850371, 8797116290975003, 4872532317019728133, 1657631603843299234219, 247098748783812523360613, 77729277912104164732573547, 1503342018433974345747514544039 (list; graph; refs; listen; history; text; internal format)



For small angles the period T of a simple gravity pendulum obeys Christiaan Huygens’s law, i.e. T = 2*Pi*sqrt(L/g) with L the length of the pendulum and g the acceleration due to gravity. For arbitrary amplitudes the period T is given below, see Wikipedia. The Taylor series expansion of T as a function of the angular displacement phi leads for the numerators of the even powers of phi to the sequence given above and for the denominators to A223068.


C. D. Andriesse and Sally Miedema, Huygens: The Man Behind the Principle, Ch. 8, 2005.


Table of n, a(n) for n=0..14.

Wikipedia, Pendulum and Pendulum mathematics.


T = 2*Pi*sqrt(L/g)*(2/Pi)*K(sin(phi/2)) with K(k) the complete elliptic integral of the first kind.

T = 2*Pi*sqrt(L/g)/M(1,cos(phi/2)) where M(x,y) = (Pi/4)*((x+y)/(K((x-y)/(x+y)) is the arithmetic-geometric mean of x and y. - Johannes W. Meijer, Dec 28 2016

Let S = Sum_{n>=0} (-1)^n*euler(2*n)*x^n/(2*n) then a(n) = numerator(1/(2*n)! * [x^n] exp(S)). - Peter Luschny, Jan 05 2017


T = 2*Pi*sqrt(L/g) * (1 + (1/16)*phi^2 + (11/3072)*phi^4 + (173/737280)*phi^6 + … ).


nmax:=14: f := series((2/Pi)*EllipticK(sin(phi/2)), phi, 2*nmax+1): for n from 0 to nmax do a(n):= numer(coeff(f, phi, 2*n)) od: seq(a(n), n=0..nmax); # End first program.

nmax:=14: f := series(1/((Pi/4)*(1+cos(phi/2))/EllipticK((1-cos(phi/2))/(1+cos(phi/2)))), phi, 2*nmax+1): for n from 0 to nmax do a(n):= numer(coeff(f, phi, 2*n)) od: seq(a(n), n=0..nmax); # End second program. - Johannes W. Meijer, Dec 28 2016


s = Series[EllipticK[Sin[t/2]^2 ], {t, 0, 60}]; CoefficientList[s/Pi, t^2] // Numerator (* Jean-François Alcover, Oct 07 2014 *)



def A223067_list(prec):

    P.<x> = PowerSeriesRing(QQ, default_prec=2*prec)

    g = lambda x: exp(sum((-1)^k*euler_number(2*k)*x^k/(2*k) for k in (1..prec+1)))

    R = P(g(x)).coefficients()

    return [numerator(R[n]/factorial(2*n)) for n in (0..prec)]

print(A223067_list(14)) # Peter Luschny, Jan 05 2017


Cf. A223068 (denominators), A019692 (2*Pi).

Cf. A280442, A280443

Sequence in context: A133243 A230604 A161355 * A280442 A218330 A196664

Adjacent sequences:  A223064 A223065 A223066 * A223068 A223069 A223070




Johannes W. Meijer, Mar 14 2013



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Last modified September 24 21:35 EDT 2020. Contains 337322 sequences. (Running on oeis4.)