cp's OEIS Frontend

The denominators are given in A101029.

The limit s = lim_{n -> infinity} s(n) with s(n) defined below equals 3*Sum_{k >= 1} zeta(2*k+1)/2^(2*k) with Euler's (or Riemann's) zeta function. This limit is 3*(2*log(2)-1) = 1.158883083...; see the Abramowitz-Stegun reference p. 259, eq. 6.3.15 with z = 1/2 together with p. 258, eqs. 6.3.5 and 6.3.3.

This is the first member (m = 2) of a family of rational partial sum sequences s(n,m) = (m-1)*m*(m+1)*Sum_{k = 1..n} 1/((m*k-1)*(m*k)*(m*k+1)) which have limit s(m) = lim_{n -> infinity} s(n,m) = -(gamma + Psi(1/m) + m/2 + Pi*cot(Pi*x)/2), with the Euler-Mascheroni constant gamma and the digamma function Psi. The same limit is reached by (m^2-1)*Sum_{k >= 0} zeta(2*k+1)/m^(2*k).

From Peter Bala, Feb 17 2022: (Start)

Let F(n) = (6*n/(2*n-1))*( 1/(1*2)*(n-1)/n - 1/(2*3)*(n-1)*(n-2)/(n*(n+1)) + 1/(3*4)*(n-1)*(n-2)*(n-3)/(n*(n+1)*(n+2)) - 1/(4*5)*(n-1)*(n-2)*(n-3)*(n-4)/(n*(n+1)*(n+2)*(n+3)) + ...). Then F(n+1) = 6*Sum_{k = 1..n} 1/((2*k-1)*(2*k)*(2*k+1)). Cf. A082687.

This identity allows us to extend the definition of Sum_{k = 1..n} 1/((2*k-1)*(2*k)*(2*k+1)) to non-integral values of n. (End)

Agrees largely with A101029. The first difference is at n=8 for which a(8)/A101029(8) = 5. There are also other differences; the ratio of the two series entries appears to be always an integer.