Given a rectangular \(m\) vertex by \(n\) vertex grid, let \(f(m,n)\) be the number of Hamiltonian paths from the lower left corner (LL) to the upper right corner (UR). The paper gives a generating function which is the quotient of two polynomials, and contains the sequence \(f(m,1)\), \(f(m,2)\), \(f(m,3)\), \(f(m,4),\dots\) as the coefficients in its Taylor series expansion about 0. It is shown that for \(m=3\) the grid has \(2^{n-2}\) Hamiltonian paths from its LL corner to its UR corner and from its LL corner to its LR corner.