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 $i \geq 1$. 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.

中文翻译：

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Wythoff 序列的和集、斐波那契表示等

令 $\alpha = (1+\sqrt{5})/2$ 并通过 $a_i = \lfloor i \alpha \rfloor$, $b_i = \lfloor i \alpha^2 \rfloor 定义上下 Wythoff 序列$ 为 $i \geq 1$。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。