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A130035
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Numerators of partial sums of a series for the inverse of the arithmetic-geometric mean (agM) of sqrt(3)/2 and 1.
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3
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1, 17, 1097, 17577, 4500937, 72018961, 4609266865, 73748453881, 75518458183369, 1208295478677929, 77330912768811177, 1237294612076514873, 316747421148616537009, 5067958740068059597769, 324349359389501776687841
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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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.
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REFERENCES
| D. A. Cox, The arithmetic-geometric mean of Gauss, L'Enseignement Math\'ematique 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
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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.
W. Lang, Rationals and limit.
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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.
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CROSSREFS
| Cf. A129934/A130034 rationals for 90 degrees deflection angle.
Sequence in context: A046731 A179157 A130449 * A032629 A075602 A172456
Adjacent sequences: A130032 A130033 A130034 * A130036 A130037 A130038
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KEYWORD
| nonn,frac,easy
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Jun 01 2007
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