In this paper, a family of weighted neighborhood sequence distance functions defined on the square grid is presented. With this distance function, the allowed weight between any two adjacent pixels along a path is given by a weight sequence. We build on our previous results, where only two or three unique weights are considered, and present a framework that allows any number of weights. We show that the rotational dependency can be very low when as few as three or four unique weights are used. An algorithm for computing the distance transform (DT) that can be used for image processing applications is also presented.