Structures in Additive Sequences

TL;DR

This paper studies the structure of a greedy additive sequence, revealing conditions for the number of even terms, its eventual pattern as arithmetic progressions, and connections to the Sierpinski Triangle.

Contribution

It characterizes the parity structure of the sequence $ ext{V}(2,n)$ and shows its eventual union of arithmetic progressions, linking it to fractal geometry and existing sequence families.

Findings

Sequence $ ext{V}(2,n)$ has exactly two even terms if $n-1$ is not a power of 2.

When $n-1$ is not a power of 2, the sequence eventually forms arithmetic progressions.

Connection established between $ ext{V}(2,n)$ and the Sierpinski Triangle.

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.