0,2

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.

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

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, December 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, December 1972, p. 591, 17.3.11.

W. Lang, Rationals and limit.

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.

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

Sequence in context: A053935 A086699 A027834 this_sequence A003303 A012838 A061685

Adjacent sequences: A129931 A129932 A129933 this_sequence A129935 A129936 A129937

nonn,frac,easy

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Jun 01 2007