We adopt the notation of the continued fraction, for $a_i,b_i\in\mathbb Z_{>0}$, $$\dfrac{a_1}{b_1\pm}\dfrac{a_2}{b_2\pm}\dfrac{a_3}{b_3\pm}\dots:=\dfrac{a_1}{b_1\pm\dfrac{a_2}{b_2\pm\dfrac{a_3}{b_3\pm\dots}}}.$$ Then $$\dfrac{\pi^2}{6}=\dfrac{1}{0^2+1^2-}\dfrac{1^4}{1^2+2^2-}\dfrac{2^4}{2^2+3^2-}\dots.$$
Is this show that $\frac{\pi^2}{6}$ is irrational? Since the pattern in the fraction is not repeated. Or is there any generalized continued fraction that has no repeated pattern and is rational? For example we have the repeated rational$$1=\dfrac{1}{2-}\dfrac{2}{3-}\dfrac{2}{3-}\dots.$$
But I think it's difficult to find the counter example for that non repeated pattern rational number and I have no clue to the theorem I guessed. (if the above is true.)
Edited: As Alex's comments. What if we add the condition $\gcd(a_i,b_{i+1})=1$ for infinitely many $i$. Is there still have any counter example?