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：我们发展了加权汉诺塔的统一代数理论，其中移动代价为任意非负对称函数，依赖于圆盘索引和柱子。从两种竞争策略（一次最大圆盘移动（one-LDM）与两次最大圆盘移动（two-LDM））的一般最优性递推出发，我们推导了两种情形下的完整矩阵公式，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主导，其谱分解揭示了与雅各布斯塔尔序列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。该框架为广泛类型的权重模型提供了精确解，包括柱对称、盘对称、多项式、几何、算术及序列诱导的代价。特别地，将经典整数序列（如斐波那契数列、卢卡斯数列、雅各布斯塔尔数列、佩尔数列、欧拉数列等）作为圆盘权重，可生成具有显式公式和递推关系的新派生序列，从而将汉诺塔确立为一种序列生成变换。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，揭示了相变现象：最优策略从针对小圆盘的 two-LDM 行为，在有限阈值后切换为 one-LDM 行为。我们的结果提供了对加权汉诺塔动力学的全面代数与组合理解，并揭示了最优解与经典整数序列之间的深层联系。