We develop a unified algebraic theory of the weighted Tower of Hanoi with arbitrary nonnegative symmetric move costs depending on both disc index and pegs. Starting from a general optimality recurrence with two competing strategies -- one largest-disc move (one-LDM) and two largest-disc moves (two-LDM) -- we derive complete matrix formulations for both regimes and obtain explicit closed forms for the minimal transfer cost. The one-LDM dynamics is governed by a nontrivial linear operator whose spectral decomposition reveals a fundamental connection with the Jacobsthal and Lichtenberg sequences, while the two-LDM dynamics exhibits pure exponential growth. This framework yields exact solutions for broad classes of weight models, including peg-symmetric, disc-symmetric, polynomial, geometric, arithmetic, and sequence-induced costs. In particular, choosing classical integer sequences (Fibonacci, Lucas, Jacobsthal, Pell, Euler, etc.) as disc weights produces new derived sequences with explicit formulas and recurrences, establishing the Tower of Hanoi as a sequence-generating transform. We further introduce and analyze models with forbidden moves and move-type-dependent weights, uncovering a phase transition phenomenon in which the optimal strategy switches from two-LDM behavior for small discs to one-LDM behavior beyond a finite threshold. Our results provide a comprehensive algebraic and combinatorial understanding of weighted Hanoi dynamics and expose deep connections between optimal solutions and classical integer sequences.

翻译：摘要：我们发展了一种统一的加权汉诺塔代数理论，该理论允许基于圆盘索引和柱子的任意非负对称移动成本。从包含两种竞争策略——一次最大圆盘移动（one-LDM）和两次最大圆盘移动（two-LDM）——的通用最优递归出发，我们推导了两种机制下的完整矩阵形式，并获得了最小转移成本的显式闭式解。one-LDM动力学由一个非平凡线性算子控制，其谱分解揭示了与雅可比斯和利希滕贝格序列的基本联系，而two-LDM动力学则呈现纯指数增长。该框架为广泛类别的权重模型提供了精确解，包括柱对称、圆盘对称、多项式、几何、算术及序列诱导成本。特别地，选择经典整数序列（如斐波那契、卢卡斯、雅可比斯、佩尔、欧拉等）作为圆盘权重会产生具有显式公式和递归的新派生序列，确立了汉诺塔作为序列生成变换的性质。我们进一步引入并分析了带有禁止移动和移动类型依赖权重的模型，揭示了一种相变现象：最优策略从小圆盘的two-LDM行为切换到大圆盘的one-LDM行为（超过有限阈值）。我们的结果为加权汉诺塔动力学提供了全面的代数和组合理解，并揭示了最优解与经典整数序列之间的深层联系。