ISL 2022 A8

For a positive integer $n$, an $n$-sequence is a sequence $\left(a_{0}, \ldots, a_{n}\right)$ of non-negative integers satisfying the following condition: if $i$ and $j$ are non-negative integers with $i+j \leqslant n$, then $a_{i}+a_{j} \leqslant n$ and $a_{a_{i}+a_{j}}=a_{i+j}$. Let $f(n)$ be the number of $n$-sequences. Prove that there exist positive real numbers $c_{1}, c_{2}$ and $\lambda$ such that $$c_{1} \lambda^{n}<f(n)<c_{2} \lambda^{n}$$ for all positive integers $n$.