MathNet 2021 — Part 05 Question 29

Let $f: \mathbb{N} \rightarrow \mathbb{N}$ be a strictly increasing function such that $f(1)=1$ and $f(2 n) f(2 n+1)=9 f(n)^2+3 f(n)$ for all $n \in \mathbb{N}$. Compute $f(137)$.