Some Formulas for Numbers of Line Segments and Lines in a Rectangular Grid

Abstract

We present a formula for the number of line segments connecting $q+1$ points of an $n_1\times \cdots \times n_k$ rectangular grid. As corollaries, we obtain formulas for the number of lines through at least $k$ points and, respectively, through exactly $k$ points of the grid. The well-known case $k=2$ is thus generalized. We also present recursive formulas for these numbers assuming $k=2,n_1=n_2$. The well-known case $q=2$ is thus generalized.