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A129934 Numerators of partial sums of a series for the inverse of the arithmetic-geometric mean (agM) of sqrt(2)/2 and 1. 3
1, 9, 297, 2401, 308553, 2472393, 79169937, 633543537, 324415700169, 2595473345377, 83057280475785, 664466019342321, 85052107504546609, 680418550231378497, 21773418753366542529, 174187444016951914257 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The denominators are found in A130034.

The rationals r(n)=a(n)/A130034(n) (in lowest terms) converge for n->infinity to 1/agM(1,sqrt(2)/2). The value for sqrt(2)/2 is approx. 0.707.

1/agM(1,sqrt(2)/2) approx. 1.180340599 multiplies 2*Pi*sqrt(l/g) to give the period T of a (mathematical) pendulum with maximal deflection of 90 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(2)/2)=(2/Pi)*K(1/2); complete elliptic integral of the first kind (see the Abramowitz-Stegun reference). K(1/2)=F(sqrt(2)/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. See 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..15.

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) = numer( sum((((2*j)!/(j!^2))^2)*(1/2^(5*j)),j=0..n)), n>=0.

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

EXAMPLE

Rationals r(n) = [1, 9/8, 297/256, 2401/2048, 308553/262144, 2472393/2097152, ...]

CROSSREFS

Sequence in context: A086699 A027834 A175823 * A003303 A012838 A216966

Adjacent sequences:  A129931 A129932 A129933 * A129935 A129936 A129937

KEYWORD

nonn,frac,easy

AUTHOR

Wolfdieter Lang, Jun 01 2007

STATUS

approved

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Last modified August 3 04:47 EDT 2021. Contains 346435 sequences. (Running on oeis4.)