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.

VARIATION ON A THEME

Now define another sequence c(n) in Z/2[x] by the recursion

c(n+6)=c(n+5)+(x^6+x^5+x^2+x)c(n)+(x^n)*(x+x^2), and the initial conditions c(0)=0, c(1)=1, c(2)=1, c(3)=x, c(4)=x^2, c(5)=x^4+x^2+x.

Question: If n is 0 or 2 mod 6, is c(n) a sum of c(k) with k less than n?

(1) I've checked that this holds through n=44.

(2) This question bears the same relation to mod 2 modular forms of level 5 that my initial question bears to level 3.