Enumeration of $k$-Fibonacci Paths using Infinite Weighted Automata

In this paper, we introduce a new family of generalized colored Motzkin paths, where horizontal steps are colored by means of $F_{k,l}$ colors, where $F_{k,l}$ is the $l$th $k$-Fibonacci number. We study the enumeration of this family according to the length. For this, we use infinite weighted automata.

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