# Does $~4^y+1=4xy^2+x~$ have infinitely many solutions in integers?

Consider the equation :

$~4^y+1=4xy^2+x$

I have found that this equation has integer solutions for following values of $~y$ :

$y\in \{1,2,193,10068,29570,..\}$

Question :

Are there finitely or infinitely many solutions ?

Can one in practice compute full list of solutions ?

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