SILK-ROAD 2005 Problem 4

Suppose $\{a(n) \}_{n=1}^{\infty}$ is a sequence that: $$a(n) =a(a(n-1))+a(n-a(n-1)) \ \ \ \forall \ n \geq 3$$ and $a(1)=a(2)=1$ .

Prove that for each $n \geq 1$ , $a(2n) \leq 2a(n)$ .