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A130037 Numerators of partial sums of a series for the inverse of the arithmetic-geometric mean (agM) of 1/2 and 1. 3
1, 19, 1297, 21427, 5584537, 90317059, 5819191945, 93509568787, 96025484363113, 1539315795453883, 98642187446349841, 1579652412024652483, 404633901283356405409, 6476837137305655553419, 414637849146342799444441 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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.

The denominators coincide with A130036.

The rationals r(n) = a(n)/A130036(n) (in lowest terms) converge for n->infinity to 1/agM(1,1/2).

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.

REFERENCES

D. A. Cox, The arithmetic-geometric mean of Gauss, L'Enseignement Mathématique, 30 (1984), 275-330. Also 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, E. M. Lifschitz: Lehrbuch der Theoretischen Physik, Band I, Mechanik, p. 30

LINKS

Table of n, a(n) for n=0..14.

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.

W. Lang, Rationals and limit

FORMULA

a(n) = numerator(Sum_{j=0..n} ((2*j)!/(j!^2))^2*((3/2^6)^j)), n >= 0.

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.

CROSSREFS

Cf. A130035/A130036 rationals for deflection angle of 60 degrees.

Sequence in context: A253127 A078955 A107673 * A047910 A237429 A177611

Adjacent sequences:  A130034 A130035 A130036 * A130038 A130039 A130040

KEYWORD

nonn,frac,easy

AUTHOR

Wolfdieter Lang, Jun 01 2007

STATUS

approved

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Last modified July 29 09:41 EDT 2021. Contains 346344 sequences. (Running on oeis4.)