International Mathematical Olympiad 1976 Problem 6

A sequence $\{u_n\}$ is defined by $$u_0 = 2, \quad u_1 = 5/2, \quad u_{n+1} = u_n(u_{n-1}^2 - 2) - u_1 \text{ for } n = 1, 2, \cdots$$

Prove that for positive integers $n$, $$[u_n] = 2^{[2^n - (-1)^n]/3}$$ where $[x]$ denotes the greatest integer $\leq x$.