Finding all paths with lengths in a fixed interval in sparse graphs
Assuming a directed graph.

Paths of length $1$ are simply the edges.

paths of length $i+1$:

Unite on all nodes $v \in V$ {concatenate all paths of length $i$ that end with $v$ with all paths of length $1$ that start with $v$}.

repeat for $i=1$ to $7$.

Note that this method avoids redundancies, so you don't need to check for them.