On the growth of Stanley sequences

Abstract

From an initial list of nonnegative integers, we form a Stanley sequence by recursively adding the smallest integer such that the list remains increasing and no three elements form an arithmetic progression. Odlyzko and Stanley conjectured that every Stanley sequence (an) satisfies one of two patterns of asymptotic growth, with no intermediate behavior possible. Sequences of Type 1 satisfy α/2≤lim infn→∞an/nlog23≤lim supn→∞an/nlog23≤α, for some constant α, while those of Type 2 satisfy an=Θ(n2/logn). In this paper, we consider the possible values for α in the growth of Type 1 Stanley sequences. Whereas Odlyzko and Stanley considered only those Type 1 sequences for which α equals 1, we show that α can in fact be any rational number that is at least 1 and for which the denominator, in lowest terms, is a power of 3.