AI 中文总结
本文针对2-和自由序列提出精确的周期性猜想，若成立则所有此类序列最终周期，并给出大量计算证据支持。
AI 中文摘要
作为先前一篇专注于$3$-和自由序列的论文的补充，本文仅考虑$2$-和自由序列：从正整数$f$和$g>f$出发，无限递增的$2$-和自由序列$S_{f,g}$按如下方式构造。在任意初始段之后，下一项是超过所有先前项且与序列中所有不同数对之和均不相同的最小正整数。根据前一篇论文中的一个定理，对于每个$f\geq 1$以及所有$f+1\leq g<2f$，序列$S_{f,g}$最终呈现周期性行为。在本文中，我们陈述了精确的猜想，若这些猜想成立，将意味着每个$2$-和自由序列都是最终周期的。这里递增序列的周期性被理解为相邻差序列的周期性，或等价地，其特征序列的周期性。我们提供了大量计算证据来支持这些猜想。
英文摘要
Complementing an earlier paper, which focused on $3$-sumfree sequences, we here consider only $2$-sumfree sequences: starting with positive integers $f$ and $g>f$, the infinite, increasing 2-sumfree sequence $S_{f,g}$ is constructed as follows. After any initial segment, the next entry is the smallest positive integer exceeding all previous ones that differs from all sums of distinct pairs in the sequence. It follows from a theorem in the previous paper that for every $f\geq 1$ and all $f+1\leq g<2f$ the sequence $S_{f,g}$ exhibits ultimately periodic behaviour. In this paper we state precise conjectures that, if true, would imply that every $2$-sumfree sequence is ultimately periodic. Here periodicity of an increasing sequence is understood to mean periodicity of the sequence of first differences, or, equivalently, of its characteristic sequence. We supply much computational evidence to support the conjectures.
Comments15 pages
