On the growth of Stanley sequences

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 ( a n ) satisfies one of two patterns of asymptotic growth, with no intermediate behavior possible. Sequences of Type 1 satisfy α / 2 ? lim inf n ? ∞ a n / n log 2 3 ? lim sup n ? ∞ a n / n log 2 3 ? α , for some constant α , while those of Type 2 satisfy a n = ? ( n 2 / log n ) . 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.