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