<s>[INST] How to prove this determinant is $\pi$? 
prove or disprove
  $$\pi=\begin{vmatrix}
3&1&0&0&0&\cdots\\
-1&6&1&0&0&\cdots\\
0&-1&\dfrac{6}{3^2}&1&0&\cdots\\
0&0&-1&\dfrac{3^2\cdot 6}{5^2}&1&\cdots\\
0&0&0&-1&\dfrac{5^2\cdot 6}{3^2\cdot 7^2}&\cdots\\
0&0&0&0&-1&\dfrac{3^2\cdot 7^2\cdot 6}{5^2\cdot 9^2}&\cdots\\
0&0&0&0&0&-1&\dfrac{5^2\cdot 9^2\cdot 6}{3^2\cdot 7^2\cdot 11^2}&\cdots\\
\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\ddots\\
\end{vmatrix}$$

I found maybe this is true.and is very interesting,(It seems Euler proved it?),because this  follows from the Euler result:
$$\pi=3+\dfrac{1^2}{6+\dfrac{3^2}{6+\dfrac{5^2}{6+\dfrac{7^2}{6+\cdots}}}}$$
and can we solve it? Thank you 
 [/INST] A proof can be found on pages 11-13 of An Elegant Continued Fraction for $\pi$. Try deriving from the Leibnitz formula $1 - \frac{1}{3} + \frac{1}{5} - ... = \frac{\pi}{4}$
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