  An integer sequence is said to be 3-free if no three elements form an
arithmetic progression. Following the greedy algorithm, the Stanley sequence
$S(a_0,a_1,\ldots,a_k)$ is defined to be the 3-free sequence $\{a_n\}$ having
initial terms $a_0,a_1,\ldots,a_k$ and with each subsequent term $a_n>a_{n-1}$
chosen minimally such that the 3-free condition is not violated. Odlyzko and
Stanley conjectured that Stanley sequences divide into two classes based on
asymptotic growth patterns, with one class of highly structured sequences
satisfying $a_n\approx \Theta(n^{\log_2 3})$ and another class of seemingly
chaotic sequences obeying $a_n=\Theta(n^2/\log n)$. We propose a rigorous
definition of regularity in Stanley sequences based on local structure rather
than asymptotic behavior and show that our definition implies the corresponding
asymptotic property proposed by Odlyzko and Stanley. We then construct many
classes of regular Stanley sequences, which include as special cases all such
sequences previously identified. We show how two regular sequences may be
combined into another regular sequence, and how parts of a Stanley sequence may
be translated while preserving regularity. Finally, we demonstrate that certain
Stanley sequences possess proper subsets that are also Stanley sequences, a
situation that appears previously to have been assumed impossible.
