Let c(n) in 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)*(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.