A130039 Numerators of partial sums of a series for the inverse of the arithmetic-geometric mean (agM) of 2/sqrt(5) and 1.

1, 21, 1689, 6761, 432753, 216380469, 17310490881, 346210001661, 88629768707061, 70903816147601, 709038163609433721, 14180763279964210461, 4537844250045576077041, 18151377000520343309289

Offset

0,2

Comments

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.

References

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.

Links

Table of n, a(n) for n=0..13.

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].

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

D. A. Cox, The arithmetic-geometric mean of Gauss, L'Enseignement Mathématique, 30 (1984), 275-330.

Wolfdieter Lang, Rationals and limit.

Formula

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.

Crossrefs

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

Sequence in context: A202800 A203361 A192843 * A221033 A221497 A155740

Adjacent sequences: A130036 A130037 A130038 * A130040 A130041 A130042

Keyword

nonn,frac,easy

Author

Wolfdieter Lang, Jun 01 2007

Status

approved