Arithmetic Progressions, Quote:
"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?
Best Answer
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 $3$.