|
%I #29 Jan 07 2024 08:16:23
%S 1,19,1297,21427,5584537,90317059,5819191945,93509568787,
%T 96025484363113,1539315795453883,98642187446349841,
%U 1579652412024652483,404633901283356405409,6476837137305655553419,414637849146342799444441
%N Numerators of partial sums of a series for the inverse of the arithmetic-geometric mean (agM) of 1/2 and 1.
%C 1/agM(1,1/2) approx. 1.372880501 multiplies 2*Pi*sqrt(l/g) to give the period T of a (mathematical) pendulum on a massless stiff wire of length l with maximal deflection of 120 degrees from the downward vertical. The gravitational acceleration on the earth's surface is g approx. 9.80665 m/s^2.
%C The denominators coincide with A130036.
%C The rationals r(n) = a(n)/A130036(n) (in lowest terms) converge for n->infinity to 1/agM(1,1/2).
%C 1/agM(1,1/2) = (2/Pi)*K(3/4); complete elliptic integral of the first kind (see the Abramowitz-Stegun reference). K(3/4) = F(sqrt(3)/2,Pi/2) in the Cox reference.
%D D. A. Cox, The arithmetic-geometric mean of Gauss, in L. Berggren, J, Borwein, P. Borwein, Pi: A Source Book, Springer, 1997, pp. 481-536. eqs.(1.8) and (1.9).
%D L. D. Landau and E. M. Lifschitz: Lehrbuch der Theoretischen Physik, Band I, Mechanik, p. 30.
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 591, 17.3.11.
%H D. A. Cox, <a href="https://doi.org/10.5169/seals-53831">The arithmetic-geometric mean of Gauss</a>, L'Enseignement Mathématique, 30 (1984), 275-330.
%H Wolfdieter Lang, <a href="/A130037/a130037.txt">Rationals and limit</a>
%F a(n) = numerator(Sum_{j=0..n} ((2*j)!/(j!^2))^2*((3/2^6)^j)), n >= 0.
%F a(n) = numerator(1 + Sum_{j=1..n} ((2*j-1)!!/(2*j)!!)^2*(3/4)^j), n >= 0, with the double factorials A001147 and A000165.
%Y Cf. A130035/A130036 rationals for deflection angle of 60 degrees.
%K nonn,frac,easy
%O 0,2
%A _Wolfdieter Lang_, Jun 01 2007
|
