A130039 - OEIS

%I #28 Jan 07 2024 08:26:51

%S 1,21,1689,6761,432753,216380469,17310490881,346210001661,

%T 88629768707061,70903816147601,709038163609433721,

%U 14180763279964210461,4537844250045576077041,18151377000520343309289

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

%C The denominators are found in A130040.

%C 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.

%C 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)).

%C 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 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).

%D L. D. Landau and E. M. Lifschitz: Lehrbuch der Theoretischen Physik, Band I, Mechanik, p. 30.

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 591, 17.3.11.

%H D. A. Cox, <a href="https://doi.org/10.5169/seals-53831">The arithmetic-geometric mean of Gauss</a>, L'Enseignement Mathématique, 30 (1984), 275-330.

%H Wolfdieter Lang, <a href="/A130037/a130037.txt">Rationals and limit.</a>

%F a(n) = numer(sum((((2*j)!/(j!^2))^2) *((1/(5*2^4))^j),j=0..n)), n>=0.

%F 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.

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

%K nonn,frac,easy

%O 0,2

%A _Wolfdieter Lang_, Jun 01 2007