How to find its closed-form? here is a sequence defined by the below recursion formula:
$$a_n=2a_{n-1}+a_{n-2}$$
where $n \in \mathbb{N}$ and $a_0=1,a_1=2$.how to find its closed-form.

To put it simple: Use a power trial function $a_n=q^n$ to find a solution disregarding the initial conditions (there will be two solutions $q_1$ and $q_2$ that can work). Next make a linear combination $Aq_1^n+Bq_2^n$ to match the initial conditions. Note that a linear combination is automatically also a solution.
