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.

EDIT-- I have further questions related to the initial recursion:

Let T: Z/2[x]-->Z/2[x] be the Z/2-linear map taking x^n to the c(n) of my initial recursion. T is degree lowering; in particular if q is a power of 2, T acts nilpotently on the space V(q) of polynomials of degree less than or equal to (q^2+2)/3, and its matrix decomposes into Jordan blocks. When q is at most 16, it appears that the largest block has size q+1.

Question: Is this true for all q?

1. Note that if my initial conjecture holds, then (when q>1) the number of Jordan blocks is (q^2+8)/12. It would interest me to know what the sizes of all these Jordan blocks are--does the computer give a plausible suggestion?

2. The map T can be interpreted as U_3+I acting on the space of odd mod 2 modular forms of level Gamma_0 (3), so the underlying question concerns the nilpotence order of such forms under the action of U_3+I. This is the basic motivation.