Školjka

%V0 A sequence $(u_{n})$ is defined by $$u_{0}=2 \quad u_{1}=\frac{5}{2}, u_{n+1}=u_{n}(u_{n-1}^{2}-2)-u_{1} \quad \textnormal{for } n=1,\ldots$$ Prove that for any positive integer $n$ we have $$[u_{n}]=2^{\frac{(2^{n}-(-1)^{n})}{3}}$$(where [x] denotes the smallest integer $\leq$ x)$.$