发表机构
Hebei Normal University; University of Bucharest; Simion Stoilow Institute of Mathematics of the Romanian Academy（河北师范大学; 布加勒斯特大学; 罗马尼亚科学院西蒙·斯托伊洛数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究三个递归二维序列，用三或四个字母模拟有限序数的冯·诺依曼构造，显式描述单元和矩形嵌套，证明组装规则，并推导嵌套盒子计数公式，同时证明这些序列非$k$-自动。
AI 中文摘要
我们研究了三个递归双序列，其彩色图编码了有限序数构造的镜像版本。两个序列使用三个字母，并实现了具有两个和三个初始符号的变体。一个四字母序列从单个初始符号实现了冯·诺依曼构造。对于每个序列，我们显式地描述其单元，并识别其矩形的有序嵌套。对于四字母的情况，我们区分了普通盒子和虚拟盒子，并证明了它们的组装规则。证明沿反对角线进行归纳。对于具有一个和三个初始符号的构造，我们将递归归结为有限集的局部构型，并通过计算机验证这些集合。我们推导了嵌套盒子数量的公式，包括算术项表示，并证明了这三个序列中任何一个都不是对任何整数$k \geq 2$的$k$-自动序列。
英文摘要
We study three recurrent double sequences whose colored diagrams encode mirrored versions of the finite-ordinal construction. Two sequences use three letters and realize variants with two and three initial symbols. A four-letter sequence realizes the von Neumann construction from one initial symbol. For each sequence we describe its cells explicitly and identify the ordered nesting of its rectangles. For four letters, we distinguish ordinary and virtual boxes and prove their assembly rules. The proofs proceed by induction along antidiagonals. For the constructions with one and three initial symbols, we reduce the recurrence to finite sets of local configurations and verify those sets by computer. We derive formulas for the numbers of nested boxes, including arithmetic-term representations, and we prove that none of the three sequences is $k$-automatic for any integer $k \geq 2$.
Comments51 pages, 6 figures
