The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
The OEIS is supported by the many generous donors to the OEIS Foundation.


(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A130035 Numerators of partial sums of a series for the inverse of the arithmetic-geometric mean (AGM) of sqrt(3)/2 and 1. 3
1, 17, 1097, 17577, 4500937, 72018961, 4609266865, 73748453881, 75518458183369, 1208295478677929, 77330912768811177, 1237294612076514873, 316747421148616537009, 5067958740068059597769, 324349359389501776687841 (list; graph; refs; listen; history; text; internal format)
The denominators are found in A130036.
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.
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.
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. 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).
L. D. Landau and E. M. Lifschitz: Lehrbuch der Theoretischen Physik, Band I, Mechanik, p. 30.
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 591, 17.3.11.
D. A. Cox, The arithmetic-geometric mean of Gauss, L'Enseignement Mathématique, 30 (1984), 275-330.
Wolfdieter Lang, Rationals and limit.
a(n) = numer(sum((((2*j)!/(j!^2))^2)*(1/2^(6*j)),j=0..n)), n>=0.
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.
Cf. A129934/A130034 rationals for 90-degree deflection angle.
Sequence in context: A221268 A179157 A130449 * A032629 A232942 A075602
Wolfdieter Lang, Jun 01 2007

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 20 19:00 EDT 2024. Contains 372720 sequences. (Running on oeis4.)