A Stanley sequence is a sequence of nonnegative integers obtained by starting with a nonempty finite increasing seed containing
no nonconstant three-term arithmetic progression,
then repeatedly adjoining the least integer greater than
the previous term that creates no nonconstant three-term arithmetic
progression (Odlyzko and Stanley 1978).
For the seed 0, 1, the terms are 0, 1, 3, 4, 9, 10, 12, 13, ... (OEIS A005836). They are precisely the nonnegative integers
whose ternary expansions use only 0 and 1. The seed 0,
4 instead gives 0, 4, 5, 7, 11, 12, 16, 23, 26, ... (OEIS A005487).
Different seeds can produce very different structures, and questions about their
growth remain open (Erdős et al. 1999, Moy and Rolnick 2016).
Questions about explicit descriptions and growth for seeds 0, form problem 271
in the Erdős problems collection (Bloom 2026).
Bloom, T. F. "Erdős Problem 271." Accessed Oct. 26, 2026. https://www.erdosproblems.com/271.Erdős,
P.; Lev, V.; Rauzy, G.; Sándor, C.; and Sárközy, A. "Greedy
Algorithm, Arithmetic Progressions, Subset Sums and Divisibility." Disc.
Math.200, 119-135, 1999. https://doi.org/10.1016/S0012-365X(98)00385-9.Moy,
R. A. and Rolnick, D. "Novel Structures in Stanley Sequences." Disc.
Math.339, 689-698, 2016. https://doi.org/10.1016/j.disc.2015.10.017.Odlyzko,
A. M. and Stanley, R. P. "Some Curious Sequences Constructed with
the Greedy Algorithm." Bell Laboratories memorandum, Jan. 1978. https://math.mit.edu/~rstan/papers/od.pdf.Sloane,
N. J. A. Sequences A005836 and A005487in "The On-Line Encyclopedia of Integer
Sequences."
form problem 271
in the Erdős problems collection (Bloom 2026).
