Let be $G=(V,E)$, where $V=\{1,\ldots,n\}$ and $E=\{\{i,j\}\subset V;|i-j|\leq k\}$ and $k<n$.  

For which values of $k\geq 2$, can we count explicitly the number of Hamiltonian paths in $G$ ? 



How many Hamiltonians Paths there are in almost regular graph ?
S. Kitaev defines Path schemes $P(n,M)$ as graphs with vertex set $\{1,2,\dots,n\}$ and edges $(i,j)$ iff $|i-j|\in M$. Hamiltonian graphs on path schemes were mentioned in "On uniquely k-determined permutations" by S. Avgustinovich and S. Kitaev. The formula is not simple even in the case where $M=\{1,2\}$ ([here](http://oeis.org/A069241)), but I guess it depends on what kind of formula you are looking for.


