is a nonaveraging sequence if it contains no three terms which are in an arithmetic
progression, i.e., terms such that

(2)

for distinct ,
,
.
The empty set and sets of length one are therefore trivially
nonaveraging.

Consider all possible subsets on the integers . There is one nonaveraging sequence on (), two on ( and ), four on , and so on. For example, 13 of the 16 subjects of are nonaveraging, with , , and excluded. The numbers of nonaveraging subsets on , , ... are 1, 2, 4, 7, 13, 23, 40, ... (OEIS A051013).

Wróblewski (1984) showed that for infinite nonaveraging sequences,

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