A characteristic 2 recurrence related to $U_3$, with a Hecke algebra application

I begin with a simple modular form motivated proof of the following: Let $C_n$ in $Z\mathrm{/}2[[t]]$ be defined by $C_{n+4} = C_{n+3} + (t^{4}+t^{3}+t^{2}+t)C_{n} + t^{n}(t^{2}+t)$,withinitialvalues$0$,$1$,$t$and$t^{2}$for$C_{0}$,$C_{1}$,$C_{2}$and$C_{3}$\mathrm{.}Thenevery$C_{4m}$ is a sum of $C_k$ with $k<4m$. This, combined with earlier results, yields: If $K$ consists of all mod $2$ modular forms of level ${\mathrm{\Gamma }}_0(3)$ annihilated by $U_2$ and $U_3+I$, then $K$ has a basis adapted (in the sense of Nicolas and Serre) to the Hecke operators $T_7$ and $T_{13}$; consequently the Hecke algebra attached to $K$ is a power series ring in these two operators.