Question 188
For a positive integer $n$, an $n$-sequence is a sequence $(a_{0}, \ldots, a_{n})$ 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
$$
c_{1} \lambda^{n}\lt f(n)\lt c_{2} \lambda^{n}
$$
for all positive integers $n$.
Let $f(n)$ be the number of $n$-sequences. Prove that there exist positive real numbers $c_{1}, c_{2}$ and $\lambda$ such that
$$
c_{1} \lambda^{n}\lt f(n)\lt c_{2} \lambda^{n}
$$
for all positive integers $n$.