TITLE: How many Hamiltonians Paths there are in almost regular graph ?
QUESTION [5 upvotes]: 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 $G$ ?

REPLY [5 votes]: Explicit values for $k\leq 9$ and small $n$ are given in the OEIS:
k=2: http://oeis.org/A003274 (contains some references and a generating function)
k=3: http://oeis.org/A174700
k=4: http://oeis.org/A174701
k=5: http://oeis.org/A174702
k=6: http://oeis.org/A177278
k=7: http://oeis.org/A177279
k=8: http://oeis.org/A177280
k=9: http://oeis.org/A177281