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A129934
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Numerators of partial sums of a series for the inverse of the arithmetic-geometric mean (agM) of sqrt(2)/2 and 1.
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3
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1, 9, 297, 2401, 308553, 2472393, 79169937, 633543537, 324415700169, 2595473345377, 83057280475785, 664466019342321, 85052107504546609, 680418550231378497, 21773418753366542529, 174187444016951914257
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OFFSET
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0,2
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COMMENTS
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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.
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REFERENCES
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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.
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LINKS
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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].
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FORMULA
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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.
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EXAMPLE
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Rationals r(n) = [1, 9/8, 297/256, 2401/2048, 308553/262144, 2472393/2097152, ...]
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CROSSREFS
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KEYWORD
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nonn,frac,easy
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AUTHOR
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STATUS
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approved
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