Recurrence relation to solve

Question

I have one problem that can be transformed into recurrence relation shown below, I guess $a_{n} = O(n)$. But I cannot solve the analytic form of $a_{n}$, can anyone help me on this? $$(n-2)a_{n+1}-(n-3)a_{n}=2a_{n-1}, n>=3$$ $$a_{2}=1, a_{3}=1$$ Follow this relation $$a_{4}=2, a_{5}=2, a_{6}=8/3, a_{7}=3$$

Answer 1

In fact, this is recurrence from problem of Misanthropic Neighbors, which can be solved by generating functions.

Misanthropic Neighbors

Limit of recursive sequence $n^2q_n=1+(n-1)^2q_{n-1}+2(n-2)q_{n-2}$