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), but I guess it depends on what kind of formula you are looking for.

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), but I guess it depends on what kind of formula you are looking for.