Consider the list of length $m$ $(1,0,\dots 0)$ we call this list $l_1$, we now define a sequence of lists recursively, where $l_1$ is the previous list, and if $l_n$ is the list $(a_1,a_2\dots a_n)$ then $l_{n+1}$ is the list $(a_1+a_2,a_2+a_3, \dots ,a_{n-1}+a_n,a_n+a_1)$ where all of the operations are done in $\mathbb Z_2$. In other words all the numbers in the lists are 1 or 0 and we use the rules $1+0=0+1=1$ and $1+1=0+0=0$.
The question is whether we can calculate what $l_n$ is for arbitrary values of $m$ and $n$.
Thank you very much in advance.
Regards.