Abstract
Let \(\alpha = (1+\sqrt{5})/2\) and define the lower and upper Wythoff sequences by \(a_i = \lfloor i \alpha \rfloor \), \(b_i = \lfloor i \alpha ^2 \rfloor \) for \(i \ge 1\). In a recent interesting paper, Kawsumarng et al. proved a number of results about numbers representable as sums of the form \(a_i + a_j\), \(b_i + b_j\), \(a_i + b_j\), and so forth. In this paper I show how to derive all of their results, using one simple idea and existing free software called Walnut. The key idea is that for each of their sumsets, there is a relatively small automaton accepting the Fibonacci representation of the numbers represented. I also show how the automaton approach can easily prove other results.
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Shallit, J. Sumsets of Wythoff sequences, Fibonacci representation, and beyond. Period Math Hung 84, 37–46 (2022). https://doi.org/10.1007/s10998-021-00390-1
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DOI: https://doi.org/10.1007/s10998-021-00390-1