Motivated by the study of Fibonacci-like Wang shifts, we define a numeration system for $mathbb{Z}$ and $mathbb{Z}^2$ based on the binary alphabet ${0,1}$. We introduce a set of 16 Wang tiles that admits a valid tiling of the plane described by a deterministic finite automaton taking as input the representation of a position $(m,n)inmathbb{Z}^2$ and outputting a Wang tile.
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