Recurrence relation sequence (Hard)

Question

I was wondering if this sequence was possible using two base cases? The sequence is $\{-1,0,1,3,13\}$ .. Ive came close by doing $(a_{n-1})^2+a_{n-2}+a_n-1$ which works for everything but the $1$. I'm jumping around the right answer but can't seem to get it.