Closed form expressions for the iterated floor function
Discrete Mathematics
Automatic Sequences: Theory, Applications, Generalizations
Automatic Sequences: Theory, Applications, Generalizations
Complexity, appeal and challenges of combinatorial games
Theoretical Computer Science - Algorithmic combinatorial game theory
Extensions and restrictions of Wythoff's game preserving its P positions
Journal of Combinatorial Theory Series A
Numeration systems: a link between number theory and formal language theory
DLT'10 Proceedings of the 14th international conference on Developments in language theory
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Let $\varphi=(1+\sqrt{5})/2$ denote the golden section. We investigate relationships between unbounded iterations of the floor function applied to various combinations of $\varphi$ and $\varphi^2$. We use them to formulate an algebraic polynomial-time winning strategy for a new four-pile take-away game Flora, which is motivated by partitioning the set of games into subsets CompGames and PrimGames. We further formulate recursive, arithmetic, and word-mapping winning strategies for it. The arithmetic one is based on the Fibonacci numeration system. We further show how to generate the floor words induced by the iterations using word-mappings and characterize them using the Fibonacci numeration system. We also exhibit an infinite array of such sequences.