Generalized Nonaveraging Integer Sequences

TL;DR

This paper extends the understanding of integer sequences avoiding certain average-based patterns, providing closed-form formulas for general m and exploring growth bounds, with computational generalizations for weighted averages.

Contribution

It generalizes the closed-form description of nonaveraging sequences from specific cases to any integer m ≥ 3 and introduces computational methods for weighted average avoidance.

Findings

Closed-form formulas for sequences avoiding averages for all m ≥ 3.

Generalization to sequences avoiding solutions to weighted averages.

Bounds established for the growth rates of these sequences.

Abstract

Let the sequence S_m of nonnegative integers be generated by the following conditions: Set the first term a_0 = 0, and for all k \geq 0, let a_k+1 be the least integer greater than a_k such that no element of {a_0,...,a_k+1} is the average of m - 1 distinct other elements. Szekeres gave a closed-form description of S_3 in 1936, and Layman provided a similar description for S_4 in 1999. We first find closed forms for some similar greedy sequences that avoid averages in terms not all the same. Then, we extend the closed-form description of S_m from the known cases when m = 3 and m = 4 to any integer m \geq 3. With the help of a computer, we also generalize this to sequences that avoid solutions to specific weighted averages in distinct terms. Finally, from the closed forms of these sequences, we find bounds for their growth rates.