Commutative algebra – a question about the duration of a sequence of polynomials satisfying a linear recurrence

Let F be a finite field and A (n) in F
— Example 1: F = Z / 2. ## EQU1 ##. The initial values ​​are: $ A (0) = A (1) = A (2) = A (3) = A (4) = 0 $, $ A (5) = $ 1.

— Example 2: F = Z / 2. $ A (n + 4) = t ^ 4A (n) + t ^ 2A (n + 1) + tA (n + 2) $. Initial values: $ A (0) = A (1) = 0, A (2) = 1, A (3) = t $.

Example 1 derives from the theory of elliptic modular forms of level 1 on Z / 2. The A (n) do span – there is a proof using the theory of the Hecke T5 operator on this space of forms, due to Nicolas and Serre, which also involves a lot of complicated calculations. (There is also a basic evidence not as complicated as I concocted).

I suppose that the A (n) of the ad hoc example 2 also extends, even if there is no underlying Hecke theory. But I do not know how to handle such issues in general. To illustrate the difficulties, note that in Example 2, the smallest n such that $ t ^ {43} $ is a sum of A (k) with k at most n is 2192. And the smallest n such that $ t ^ {107} $ is a sum of A (k) with k at most n is 3989.