Let $c(n)$ in $\mathbb{Z}_2[x]$ be defined by the recursion $c(n+4)=c(n+3)+(x^4+x^3+x^2+x)c(n)+(x^n)\cdot(x+x^2)$, and the initial conditions $c(0)=0$, $c(1)=1$, $c(2)=x$, $c(3)=x^2$.

Question: If 4 divides $n$, is $c(n)$ a sum of $c(k)$ with $k$ less than $n$?

Remarks:

(1) I've checked that this holds up to $n=64$.

(2) The recursion may seem artificial, but it arises in studying the action of the operator $U_3$ on a space of mod $2$ modular forms of level 3. This accounts for the number theory tag.