Partitioning integers to avoid arithmetic progressions of length $3$

Question

"A finite sequence like $\{10,13,16,19\}$. This is an arithmetic progression of length $4$, of (constant) difference $3$ and an initial value $10$."

In this first finite sequence my guess is that length $4$ means the number of elements in that sequence?

However, im unable to understand whats going on in this sequence:

$\{0, 1, 2, 4, 7, 11, 17, 26, 40, 61, 92, 139, 209, 314, 472, 709, 1064, 1597,...\}$. OEIS: A006999 (Partitioning integers to avoid arithmetic progressions of length 3.)

In this sequence, it is not clear what it means for me. The terms does not seem to match a constant difference pattern? And so avoiding length $3$ progressions gets me confused, Does it have anything to do with the partitioning of integers, but how does it work?

How does this avoid AP's of length $3$ when there are $0,1,2$, then $0,2,4$ and $1,4,7$? — Oscar Lanzi, Dec 31 '19 at 01:49
Not sure I follow you. The sequence contains those progressions do what is meant by avoiding three terms in AP? — Oscar Lanzi, Dec 31 '19 at 02:04

score 0 · Accepted Answer · answered Dec 31 '19 at 02:08

This entry of the OEIS could use some clarification. However, you can still figure out how to make sense of it by looking at some of the linked sequences and references. In particular, look instead at the linked entry A006997, which is discussed in the paper Greedily Partitioning the Natural Numbers into Sets Free of Arithmetic Progressions.

The sequence A006999 which you posted about is discussed at the end of this paper in the context of studying the asymptotic behavior of A006997 by looking focusing on terms of index $3^n - 1$. Meanwhile, the paper makes explicit how A006997 characterizes the process of applying a greedy algorithm to partition the natural numbers into sets that avoid arithmetic progressions of length $3$.

Before I accept your answer, what does length of $3$ mean in this case? — , Dec 31 '19 at 02:50
A set of naturals numbers $S$ contains an arithmetic progression of length $k$ if there exist distinct $a_1,\ldots,a_k \in S$ that, when put in sequence, form an arithmetic progression in the same sense as you're using in the OP. — Badam Baplan, Dec 31 '19 at 03:19

Partitioning integers to avoid arithmetic progressions of length $3$

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