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Structures in additive sequences

Volume 186 / 2018

Borys Kuca Acta Arithmetica 186 (2018), 273-300 MSC: Primary 11B13; Secondary 11B83. DOI: 10.4064/aa180425-14-8 Published online: 5 November 2018


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 $n \geq 5$ is odd, then the sequence $\mathcal{V}(2,n)$ has exactly two even terms $\{2,2n\}$ 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 the Sierpiński 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.


  • Borys KucaSchool of Mathematics
    The University of Manchester
    Alan Turing Building
    Manchester 13 9PL, UK

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