No such sequence exists. It suffices to only look at non-negative sequences. Notice $a_{n+2}<\max(a_n,a_{n+1})$. Let $M_k=\max(a_{2k-1},a_{2k})$. Then $$M_k^2\geq M_{k+1}^2+1$$This is clearly impossible as it would force $M_i^2$ to eventually become negative.