Structures in Additive Sequences

Abstract

Consider the sequence $\mathcal{V}(2,n)$ constructed in a greedy fashion by setting $a_1 = 2$, $a_2 = n$ and defining $a_{m+1}$ as the smallest integer larger than $a_m$ that can be written as the sum of two (not necessarily distinct) earlier terms in exactly one way; the sequence $\mathcal{V}(2,3)$, for example, is given by $$ \mathcal{V}(2,3) = 2,3,4,5,9,10,11,16,22,\dots$$ We prove that if $n \geqslant 5$ is odd, then the sequence $\mathcal{V}(2,n)$ has exactly two even terms $\left\{2,2n\right\}$ if and only if $n-1$ is not a power of 2. We also show that in this case, $\mathcal{V}(2,n)$ eventually becomes a union of arithmetic progressions. If $n-1$ is a power of 2, then there is at least one more even term $2n^2 + 2$ and we conjecture there are no more even terms. In the proof, we display an interesting connection between $\mathcal{V}(2,n)$ and Sierpinski Triangle. We prove several other results, discuss a series of striking phenomena and pose many problems. This relates to existing results of Finch, Schmerl & Spiegel and a classical family of sequences defined by Ulam.