By writing out the first few terms, we find that $a_{n+4}=-4 a_{n}$. Indeed, $$a_{n+4}=2\left(a_{n+3}-a_{n+2}\right)=2\left(a_{n+2}-2 a_{n+1}\right)=2\left(-2 a_{n}\right)=-4 a_{n}$$ Then, by induction, we get $a_{4 k}=(-4)^{k}$ for all positive integers $k$, and setting $k=501$ gives the answer.