Why zeta(2) in these inifinite sums?

Question

The infinite sum of the reciprocals of these two sequences have zeta(2) in the result. The value is not in OEIS. A000326 A002411 Edit---rolled back the changes. Both $\frac{1}{2}$ and $2$ are important. $$\sum _{m=1}^{\infty } \frac{1}{\sum _{n=1}^m \frac{1}{2}n (3 n-1)}= \sum _{n=1}^{\infty } \frac{1}{\frac{1}{2}n^2 (n+1)}= 2\left(\frac{\pi ^2}{6}-1\right)$$

Is there anything significant about this that might relate to zeta(2)? Or RH?

Answer 1

I should have left out the $\frac{1}{2}$ and the $2$. I would have found A011379 and this comment:

$\sum_{n>0} 1/a(n) = (\pi^2 - 6)/6 = 0.6449340..$ [Jolley eq 272] - Gary W. Adamson, Dec 22 2006

Thank-you for your efforts.