Szekeres's sequence: a(n)-1 in ternary = n-1 in binary; also: a(1) = 1, a(2) = 2, and thereafter a(n) is smallest number k which avoids any 3-term arithmetic progression in a(1), a(2), ..., a(n-1), k

Weight–level plate of Szekeres's sequence: a(n)-1 in ternary = n-1 in binary; also: a(1) = 1, a(2) = 2, and thereafter a(n) is smallest number k which avoids any 3-term arithmetic progression in a(1), a(2), ..., a(n-1), k — Click the plate to explore it in 3-D. Weight k across, level L up, both on log scales: blue in the weight class (k ≤ L), orange in the level class (k > L). The dashed diagonal is k = L.

Open in the 3-D viewer A003278 on the OEIS

Terms	100,000 (n = 1 … 100,000)
Decomposable (a > 2d)	99,983
Level class, k > L	358 · 0.36 %
Weight class, k ≤ L	99,625 · 99.64 %
Ties, k = L	17
On the level line L = 1	1
Forced level, l ≤ d²	307
Range of a(n)	1 … 57,476,669
Range of the jump d	1 … 21,523,361
Largest weight k, level L	14,348,907, 28,738,332

17 different gaps occur, from 1 to 21,523,361; the level share is 0.36 %; 17 terms do not decompose.

The decomposition

Every term of a strictly increasing sequence is written a(n) = k(n)·L(n) + d(n): the jump d = a(n+1) − a(n), the weight k the least divisor of a − d greater than d, the level L = (a − d)/k. A term decomposes when a > 2d; it is in the level class when k > L and in the weight class when k ≤ L. See decompwlj.com and arXiv:0711.0865, or how it works, with worked examples and a live one.