$a_{n-1}a_{n+1} - a_n^2 = (-1)^n$

$a_{n-1}a_{n+1} - a_n^2 = 1$

$a_{n-1}a_{n+1} - a_n^2 = (-1)^{n+1}$

$a_{n-1}a_{n+1} + a_n^2 = 0$