Consider the sequence ${a_n}: a_1 = 1, a_2 = 2,$ and $a_n = 2a_{n−1} + a_{n−2}$ for n ≥ 3. Prove that k ∈ N, $2^k | a_n$ if and only if $2^k | n$

Question

I've been working on the problem:

Consider the sequence ${a_n}: a_1 = 1, a_2 = 2,$ and $a_n = 2a_{n−1} + a_{n−2}$ for n ≥ 3. Prove that k ∈ N, $2^k | a_n$ if and only if $2^k | n$

I've programmed and found some properties:

1. $a_{n}\equiv a_{n-2^k} \pmod {2^k}$
2. $a_{2^k-1} \equiv 1 \pmod {2^k}$

I don't know how to prove them but I'm able to show that: $$\begin{aligned} a_n &= 2a_{n-1} + a_{n-2} \\ &= a_2a_{n-1}+a_1a_{n-2} \\ &=a_2(2a_{n-2}+a_{n-3})+a_1a_{n-2} \\ &=a_3a_{n-2}+a_2a_{n-3} \\ &=a_ja_{n+1-j}+a_{j-1}a_{n-j} &(2 \leq j\leq n-1) \end{aligned} $$

Is that the correct way to solve the problem?