A130035 - OEIS

%I #30 Jan 07 2024 08:52:07

%S 1,17,1097,17577,4500937,72018961,4609266865,73748453881,

%T 75518458183369,1208295478677929,77330912768811177,

%U 1237294612076514873,316747421148616537009,5067958740068059597769,324349359389501776687841

%N Numerators of partial sums of a series for the inverse of the arithmetic-geometric mean (AGM) of sqrt(3)/2 and 1.

%C The denominators are found in A130036.

%C The rationals r(n)=a(n)/A130036(n) (in lowest terms) converge for n->infinity to 1/agM(1,sqrt(3)/2). The value for sqrt(3)/2 is approx. 0.866.

%C 1/agM(1,sqrt(3)/2) approx. 1.073182007 multiplies 2*Pi*sqrt(l/g) to give the period T of a (mathematical) pendulum with maximal deflection of 60 degrees from the downward vertical. The length of the pendulum is l and g is the gravitational acceleration on the earth's surface, approx. 9.80665 m/s^2.

%C 1/agM(1,sqrt(3)/2)=(2/Pi)*K(1/4); complete elliptic integral of the first kind (see the Abramowitz-Stegun reference). K(1/4)=F(1/2,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="/A130035/a130035.txt">Rationals and limit.</a>

%F a(n) = numer(sum((((2*j)!/(j!^2))^2)*(1/2^(6*j)),j=0..n)), n>=0.

%F a(n) = numer(1+sum(((2*j-1)!!/(2*j)!!)^2*(1/4)^j,j=1..n)), n>=0, with the double factorials A001147 and A000165.

%Y Cf. A129934/A130034 rationals for 90-degree deflection angle.

%K nonn,frac,easy

%O 0,2

%A _Wolfdieter Lang_, Jun 01 2007