Counting pseudo progressions

Abstract

An $m$-pseudo progression is an increasing list of numbers for which there are at most $m$ distinct differences between consecutive terms. This object generalizes the notion of an arithmetic progression. We give two counts for the number of $k$-term $m$-pseudo progressions in $\left\{1,2,\dots ,n\right\}$. We also provide computer-generated tables of values which agree with both counts and graphs that display the growth rates of these functions. Finally, we present a generating function which counts $k$-term progressions in $\left\{1,2,\dots ,n\right\}$ whose differences are all distinct, and we discuss further directions in Ramsey theory.