0,2

1/agM(1,1/2) approx. 1.372880501 multiplies 2*Pi*sqrt(l/g) to give the period T of a (mathematical) pendulum on a massless stiff wire of length l with maximal deflection of 120 degrees from the downward vertical. The gravitational acceleration on the earth's surface is g approx. 9.80665 m/s^2.

The denominators coincide with A130036.

The rationals r(n)=a(n)/A130036(n) (in lowest terms) converge for n->infinity to 1/agM(1,1/2).

1/agM(1,1/2)=(2/Pi)*K(3/4); complete elliptic integral of the first kind (see the Abramowitz-Stegun reference). K(3/4)=F(sqrt(3)/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) *((3/2^6)^j),j=0..n)), n>=0.

a(n) = numer(1+sum(((2*j-1)!!/(2*j)!!)^2*(3/4)^j,j=1..n)), n>=0, with the double factorials A001147 and A000165.

Cf. A130035/A130036 rationals for deflection angle of 60 degrees.

Sequence in context: A091750 A078955 A107673 this_sequence A047910 A051847 A002115

Adjacent sequences: A130034 A130035 A130036 this_sequence A130038 A130039 A130040

nonn,frac,easy

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