We prove a number of conjectures recently stated by P. Barry, related to the paperfolding sequence and the Rueppel sequence. Furthermore, we study the regularity of sequences involved in the paper, and prove that for all $q\ge 2$, the sequence consisting of the positive integers whose odd part is of the form $4k+1$ is not q-regular. Finally we establish the 2-regularity of two sequences of Hankel determinants.

中文翻译：

---

关于巴里（P. Barry）的一些猜想

我们证明了P. Barry最近提出的许多猜想，它们与折纸序列和Rueppel序列有关。此外，我们研究了本文涉及的序列的规律性，并证明了对于所有$q\ge \mathrm{2个}$，由正整数组成的序列，其奇数部分的形式为 $4ķ+\mathrm{1个}$不是q-常规的。最后，我们建立了两个汉克行列式序列的2正则性。