Two fractions related to early convergents of $\pi$ are the semi-convergents $$\frac{333-22}{106-7}=\frac{311}{99}$$ and $$\frac{355+22}{113+7}=\frac{377}{120}$$
The denominators of these fractions are close to squares. $$99=10^2-1$$ $$120=11^2-1$$
This suggests looking for other approximations of $\pi$ with denominator of the form $(n^2-1)$. Among the first $1000$ cases, two prominent ones appear at $n=225$ and $n=729$.
$$ (225^2-1)\pi = (15^4-1)\pi = 50624\pi \approx 159039.986 \approx 159040$$
$$ (729^2-1)\pi = (3^{12}-1)\pi = 531440\pi \approx 1669567.9998 \approx 1669568$$
The numbers $225=15^2$ and $729=3^6$ are squares themselves and the resulting rational approximations are the fourth and sixth convergents, both from above.
$$\frac{159040}{50624}=\frac{2^6·5·7·71}{2^6·7·113}=\frac{355}{113}$$
$$\frac{1669568}{531440}= \frac{2^6·19·1373}{2^4·5·7·13·73}= \frac{104348}{33215}$$
Is this just a coincidence illusion or can it be somehow justified?