Below is a problem, from an old Silk Road olympiad.
Define an infinite sequence, $a(n)$, such that, $a(1)=a(2)=1$; $$ a(n)=a(a(n-1))+a(n-a(n-1)),\forall n\geq 3. $$ Show that, for every $n\geq 1$, $a(2n)\leq 2a(n)$.
Post Closed as "Not suitable for this site" by user6976, Anthony Quas, YCor, David Handelman, Alexey Ustinov