Let the sequence Sm of nonnegative integers be generated by the following conditions. Set the first term a0 = 0, and for all k ≥ 0, let ak+1 be the least integer greater than ak such that no element of {a0, . . . , ak+1} is the average of m ? 1 distinct other elements. Szekeres gave a closed-form description of S3 in 1936, and Layman provided a similar description for S4 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 Sm from the known cases when m = 3 and m = 4 to any integer m ≥ 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.
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