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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)
OFFSET

0,3

COMMENTS

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.

REFERENCES

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

LINKS

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

Wikipedia, Pendulum and Pendulum mathematics.

FORMULA

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

EXAMPLE

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

MAPLE

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

MATHEMATICA

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

PROG

(Sage)

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

CROSSREFS

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

KEYWORD

nonn,easy,frac

AUTHOR

Johannes W. Meijer, Mar 14 2013

STATUS

approved

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Last modified November 19 08:44 EST 2019. Contains 329318 sequences. (Running on oeis4.)