Consider an $n$-dimensional hypercubic lattice, where the set of vertices is $\mathbb{Z}^n$. Let $E$ be the union of all edges between adjacent points (i.e. the grid lines).
Given a sequence of adjacent points on the lattice, the corresponding map through $E$ is called a 'path through the lattice'. In addition, we say such a path 'has no loops' if it has a trivial path homotopy with respect to the induced topology on $E$ from $\mathbb{R}^d$.
Can we count that number of paths of length $L$ that begin and end at the origin and contain no loops?
Obviously for odd $L$ this is zero. I've also found that this number is $2n$ for $L=2$ and $2n(4n-1)$ for $L=4$. But, I'm having trouble computing even $L=6$ directly. Is there some general formula for this?
I would also be happy with an OEIS entry for some $n\geq 3$.
If I understand correctly, we can replace the $\mathbb{Z}^n$ net by the $2n$-regular tree in this problem, so this is the cogrowth sequence of a free group on $n$ generators. Check out the rows of https://oeis.org/A183135, taking even-indexed columns only.