A223067 - OEIS

A223067

A sequence related to the period T of a simple gravity pendulum for arbitrary amplitudes.

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).