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.

令\(\alpha = (1+\sqrt{5})/2\)并通过\(a_i = \lfloor i \alpha \rfloor \) , \(b_i = \lfloor i \ )定义上下 Wythoff 序列alpha ^2 \rfloor \)为\(i \ge 1\)。在最近的一篇有趣的论文中，Kawsumarng 等人。证明了一些关于可表示为\(a_i + a_j\)、\(b_i + b_j\)、\(a_i + b_j\)等形式的总和的数字的结果。在这篇论文中，我展示了如何使用一个简单的想法和名为Walnut 的现有免费软件推导出他们的所有结果. 关键思想是，对于它们的每个和集，都有一个相对较小的自动机接受所表示数字的斐波那契表示。我还展示了自动机方法如何轻松证明其他结果。