For a sequence $a_{1}<a_{2}<\cdots<a_{n}$ of integers, a pair ( $a_{i}, a_{j}$ ) 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 $n$. Please provide the value of the expression when n = 3. 

Let's think step by step and output the final answer within \boxed{}.
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