Let $c(n)$ in $\mathbb{Z}/2\mathbb{Z}[x]$ 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 $$c(0)=0,\quad c(1)=1,\quad c(2)=x,\quad c(3)=x^2.$$

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.

Now define another sequence $c(n)$ in $\mathbb Z/2\mathbb Z[x]$ by the recursion $$c(n+6)=c(n+5)+(x^6+x^5+x^2+x)c(n)+x^n\cdot(x+x^2),$$ and the initial conditions $$c(0)=0,\quad c(1)=1,\quad c(2)=1,\quad c(3)=x,\quad c(4)=x^2,\quad c(5)=x^4+x^2+x.$$

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.