AI 中文总结
本文用Walnut证明器证明了多个关于Wythoff序列、Fibonacci词和Lucas词的OEIS猜想，包括Kimberling的表示计数猜想与间隙猜想，并提供了完整命令文件。
AI 中文摘要
我们证明了整数序列在线百科全书中关于下Wythoff序列、上Wythoff序列以及Fibonacci词的若干猜想。其中包括Kimberling关于将$n=\lfloor h\varphi\rfloor+\lfloor k\varphi^2\rfloor$（其中$h,k\ge1$）写成两数之和的方式数目的三个猜想中的两个（A259598）：当且仅当$n+1=2F$（其中$F\ge2$为Fibonacci数）时，恰好存在一种表示方式；当且仅当$n+1\ge7$为Lucas数时，恰好存在两种表示方式。第三个猜想（即当且仅当$n+1$为Fibonacci数时不存在表示方式）已由Kawsumarng等人先前证明。我们还观察到，Kimberling关于A003622的两个不同项之和及其补集（A333308, A333309）的间隙的猜想，在平移$2$之后，可由Shallit以及Bosma等人关于A260317的早期Walnut结果推出，并且我们重新验证了该猜想。接下来，我们证明了Kimberling在2025年提出的关于Fibonacci词和“Lucas词”取规定值的位置之间间隙的五个猜想（A383423-A383427）。最后，我们证明了Mathar关于A285383的一个猜想，并指出Schmidt关于A003250的一个猜想可由Carlitz、Scoville和Vaughan（1973）的定理推出；我们还用Walnut对其进行了确认。大多数证明是在自由证明器Walnut中运行的决策过程，我们提供了完整的命令文件。
英文摘要
We prove several conjectures from the On-Line Encyclopedia of Integer Sequences about the lower and upper Wythoff sequences and the Fibonacci word. Among them are two of Kimberling's three conjectures on the number of ways to write $n=\lfloor hφ\rfloor+\lfloor kφ^2\rfloor$ with $h,k\ge1$ (A259598): exactly one way if and only if $n+1=2F$ for a Fibonacci number $F\ge2$, and exactly two ways if and only if $n+1\ge7$ is a Lucas number. The third conjecture, that no way exists if and only if $n+1$ is a Fibonacci number, was proved earlier by Kawsumarng et al. We also observe that Kimberling's conjecture on the gaps of the sums of two distinct terms of A003622 and of their complement (A333308, A333309) follows, after a shift by $2$, from earlier Walnut results of Shallit and of Bosma et al. on A260317, and we re-verify it. Next, we prove Kimberling's five 2025 conjectures on the gaps between positions where the Fibonacci word and the "Lucas word" take prescribed values (A383423-A383427). Finally, we prove a conjecture of Mathar on A285383, and we point out that a conjecture of Schmidt on A003250 follows from theorems of Carlitz, Scoville and Vaughan (1973); we also confirm it with Walnut. Most proofs are decision procedures run in the free prover Walnut, and we supply the complete command file.
Comments5 pages; the complete Walnut command file is an ancillary file. Also deposited at Zenodo, doi:10.5281/zenodo.23124896
