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 $n\geq 1$, $a(2n)\leq 2a(n)$.
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 $n\geq 1$, $a(2n)\leq 2a(n)$.
This is too long for a remark, but if you look at $N_m=\#\{n:a_n=m\}$, the sequence $\{N_m\}$ is
2
2
1 3
1 1 2 4
1 1 1 2 1 2 3 5
1 1 1 1 2 1 1 2 1 2 3 1 2 3 4 6 ...
To get each line from the line above it, replace each term $N$ by $1,2,\dots,N$, and then add 1 to the final term.
Not sure if this observation helps in any way.