Sequences Generated by the Lattice Path Problem with Various Vector Sets

Abstract

The lattice path problem involves finding a path between two specific points in space using only certain predefined vectors. The goal is to establish a relationship between the number of lattice paths to a point and the emergence of specific number sequences. This was achieved by analyzing lattice paths in a table within a Cartesian coordinate system. The number of paths to a particular cell, starting from the first column of the table, was computed, and the results were analyzed through a computer program. This method led to the discovery of Fibonacci, Pell and  Pell-Lucas and Tribonacci sequences. Upon examining tables in dimensions higher than two, it was observed that the numbers found corresponded to the products of these special number sequences. Recursive relations were derived for the vector sets used in this process, and through these relations, identities among the number sequences were established