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.

TWO VARIATIONS IN CHARACTERISTIC 3

Variation 2a---Let c(n) in Z/3[x] be defined by the recursion

c(n+3)=c(n+2)-(x^3+x^2+x)c(n)+x^n*(x^3-x), and the initial conditions

c(0)=0, c(1)=x, c(2)=x.

If (n,3)=1, define d(n) to be c(n)+c(n+1)+c(n+2) or c(n)-c(n+1) according as n is 1 or 2 mod 3.

Question: If n is 2 mod 9, is d(n) a Z/3-linear combo of d(k) with k less than n?

Variation 2b---Let c*(n) be c(n)-x^n with c(n) as in variation 2a. For n prime to 3 define d*(n) as in variation 2a, but with c(n) replaced by c*(n).

Question: If n is 2 mod 9, is d*(n) a Z/3 linear combo of d*(k) with k less than n?

Remarks: I've verified that these hold for n up through 83, and am confident that a variant of Peter Mueller's technique will allow one to go much further. The questions are related to calculating the kernels of U_2+I and U_2-I on the space of mod 3 modular forms of level 2, just as the earlier questions were related to calculating the kernels of U_3+I and U_5+I on the spaces of mod 2 modular forms of levels 3 and 5 respectively.

FINAL EDIT (12/7/16)

1. I've found elementary but apparently mysterious answers to the first 2 questions--see my 2 answers. The arguments are similar, and I'll randomly accept the level 3 answer.

2. I think I can answer the characteristic 3 questions too. But to avoid accusations of self-abuse I'll refrain. And though I find the question on Jordan blocks very interesting (and similar questions attached to the other recursions should also be interesting), the thread has become long and so I've deleted that question.