What is the asymptotic growth of the sequence $$a_n:=\sum_{k\geq 0} 3^k c_{n,k},$$ as $n\rightarrow\infty$, where $c_{n,k}$ denotes the number of integer compositions of $n$ with exactly $k$ many 2s?

A composition of $n$ is a sum $n=c_1+c_2+\cdots+c_p$, with all the $c_i$ positive. The first values of the sequence $a_n$ are $1,1,4,8,22,52,135,\ldots$ (not in the OEIS). [Edit: As pointed out by Somos below, the value 135 is wrong, and must be corrected to 132, and then the sequence is in the OEIS.]

So far, I was only able to prove the following bounds: As $\sum_{k\geq 0}c_{n,k}=2^{n-1}$, it follows that $$2^{n-1}\leq a_n \leq (2\sqrt{3})^n=(3.464...)^n.$$