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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)
OFFSET
0,2
COMMENTS
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.
REFERENCES
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.
LINKS
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.
FORMULA
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.
CROSSREFS
Cf. A129934/A130034 rationals for 90-degree deflection angle.
Sequence in context: A221268 A179157 A130449 * A032629 A232942 A075602
KEYWORD
nonn,frac,easy
AUTHOR
Wolfdieter Lang, Jun 01 2007
STATUS
approved

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