0,2

The denominators are found in A130040.

The rationals r(n)=a(n)/A130040(n) (in lowest terms) converge for n->infinity to 1/agM(1,2/sqrt(5)). 2/sqrt(5)= (2/5)*(-1 + 2*phi) approx. 0.894 with the golden mean phi.

1/agM(1,2/sqrt(5)) approx. 1.056549198 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 angle phi(0) of approx. 53.13 degrees from the downward vertical. The gravitational acceleration on the earth's surface is g approx. 9.80665 m/s^2. phi(0)= 2*arcsin(1/sqrt(5)).

1/agM(1,2/sqrt(5))=(2/Pi)*K(1/5); complete elliptic integral of the first kind (see the Abramowitz-Stegun reference). K(1/5)=F(1/sqrt(5),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/(5*2^4))^j),j=0..n)), n>=0.

a(n) = numer(1+sum(((2*j-1)!!/(2*j)!!)^2*(1/5)^j,j=1..n)), n>=0, with the double factorials A001147 and A000165.

Cf. A130037/A130036 rationals for deflection angle of 120 degrees.

Adjacent sequences: A130036 A130037 A130038 this_sequence A130040 A130041 A130042

Sequence in context: A130332 A035319 A081786 this_sequence A007593 A119099 A033510

nonn,frac,easy

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