Sumsets of Wythoff Sequences, Fibonacci Representation, and Beyond

06/07/2020

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by   Jeffrey Shallit, et al.

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Let α = (1+√(5))/2 and define the lower and upper Wythoff sequences by a_i = ⌊ i α⌋, b_i = ⌊ i α^2 ⌋ for i ≥ 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.