So, going in a different direction, we get a weird directed two category if we index the morphisms of the free category by length.

(Not just an enriched category of some sort because id and composition only hold while moving to a later index)

Or you can look at the graph as a binary valued matrix then take powers of it as a matrix of natural numbers, then the sequences of particular diagonals are the sequences generated by the graph pointed at that node.

So would looking at traces and determinants help with building a pointed graph that generates a particular sequence?