EGMO 2024 Problem 4

Problem. For a sequence $a_{1}<a_{2}<\cdots<a_{n}$ of integers, a pair $\left(a_{i}, a_{j}\right)$ with $1 \leq i<j \leq n$ is called interesting if there exists a pair $\left(a_{k}, a_{l}\right)$ of integers with $1 \leq k<l \leq n$ such that

$$\frac{a_{l}-a_{k}}{a_{j}-a_{i}}=2$$

For each $n \geq 3$, find the largest possible number of interesting pairs in a sequence of length $n$.