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- W2798878397 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 $left{2,2nright}$ 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." @default.
- W2798878397 created "2018-05-07" @default.
- W2798878397 creator A5067915351 @default.
- W2798878397 date "2018-04-25" @default.
- W2798878397 modified "2023-09-27" @default.
- W2798878397 title "Structures in Additive Sequences" @default.
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