AI 中文总结
本文提出旋转记忆斐波那契数，一种递推记忆周期性变化的数列，证明其刚性结构、闭式与生成函数，并通过周期平铺给出几何解释，为更高周期建模提出开放问题。
AI 中文摘要
我们引入并研究了旋转记忆斐波那契数，这是斐波那契数列的一个周期变阶模拟，其中递推中所使用的前项数量随索引循环变化。尽管记忆可变，所得序列却展现出极其刚性的结构。我们推导出闭式形式、有理生成函数、算术性质以及精确的增长行为，并证明该序列自然分解为几何子序列。我们还发展了周期约束平铺和受限组合方面的组合解释，包括对序列乘法结构的双射解释。此外，前两个非经典周期具有自然的几何驱动实现：周期2序列源于三角形链上的单体-二聚体平铺，而周期3序列与单六边形和双六边形对双六边形条带的平铺相关。这些联系为旋转递推提供了几何解释，其中周期行为由底层结构本身诱导，并提出了为更高周期构建类似模型的更广泛问题。
英文摘要
We introduce and study the rotating-memory Fibonacci numbers, a periodic variable-order analogue of the Fibonacci sequence in which the number of preceding terms used in the recurrence changes cyclically with the index. Despite this varying memory, the resulting sequences exhibit a remarkably rigid structure. We derive closed forms, rational generating functions, arithmetic properties, and exact growth behavior, and show that the sequence decomposes naturally into geometric subsequences. We also develop combinatorial interpretations in terms of periodically constrained tilings, and restricted compositions, including bijective explanations for the multiplicative structure of the sequence. In addition, the first two nonclassical periods admit natural geometry-driven realizations: the period-2 sequence arises from monomer--dimer tilings of a triangular chain, while the period-3 sequence is related to tilings of a double hexagon strip by single and double hexagons. These connections provide geometric interpretations of the rotating recurrence in which the periodic behavior is induced by the underlying structures themselves, and suggest a broader problem of constructing analogous models for higher periods.
