0,2

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.

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^(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.

Cf. A129934/A130034 rationals for 90 degrees deflection angle.

Sequence in context: A077645 A046731 A130449 this_sequence A032629 A075602 A022012

Adjacent sequences: A130032 A130033 A130034 this_sequence A130036 A130037 A130038

nonn,frac,easy

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