Finding the positive integer numbers to get $\frac{\pi ^2}{9}$ As we know, there are many formulas of $\pi$ , one of them $$\frac{\pi ^2}{6}=\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}......
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and this $$\frac{\pi ^2}{8}=\frac{1}{1^2}+\frac{1}{3^2}+\frac{1}{5^2}......$$
Now,find the positive integer numbers $(a_{0}, a_{1}, a_{2}....)$ to get $$\frac{\pi^2 }{9}=\frac{1}{a_{0}^2}+\frac{1}{a_{1}^2}+\frac{1}{a_{2}^2}....$$

$$\begin{align}\frac{\pi^2}{9}&=1+\frac{1}{3}\sum_{k=0}^\infty\frac{1}{(k+1)^2(k+2)^2}\\&=1+\frac{1}{3(1·2)^2}+\frac{1}{3(2·3)^2}+\frac{1}{3(3·4)^2}+\frac{1}{3(4·5)^2}+\ldots\\&=1+\frac{1}{12}+\frac{1}{108}+\frac{1}{432}+\frac{1}{1200}\ldots...\end{align}$$
