Consider the Penrose tiling $P3$, inflated up to $6$ generations:
We draw a line passing through the center of the tiling (red dot) and the outer vertex of the rightmost starting tile (black dot).
Some of the tiles intercepted by this line are perfectly cut in a half by it. Among these peculiar tiles, we highlight in blue the "$\color{blue}{\text{fat rhombus}}$", and in green the "$\color{green}{\text{thin rhombus}}$".
Let us focus only the peculiar tiles which are cut in a half by the line. We can consider the blue tiles as $\color{blue}0$s and the green tiles as $\color{green}1$s. Therefore, if we collect these tiles from left to right, we obtain the binary sequence
$$\color{blue} 0,\color{blue}0,\color{green}1,\color{blue}0,\color{blue}0,\color{blue}0,\color{green}1,\color{blue}0,\color{blue}0,\color{blue}0,\color{green}1,\color{blue}0,\color{blue}0,\color{green}1,\color{blue}0,\color{blue}0.$$
These $16$ terms are naturally too few for investigating the structure of the sequence. However, we can enlarge the Penrose tiling.
Therefore, we now inflate the tiling up to $8$ generations. We need an even number of generations in order to keep the same central structure and, therefore, to obtain a binary sequence that contains the previous one, the one composed of only $16$ terms.
Performing the same operation as before on the $8$-generations $P3$ tiling, we find:
which corresponds to the binary sequence
$$\color{green}1,\color{blue}0,\color{blue}0,\color{blue}0,\color{green}1,\color{blue}0,\color{blue}0,\color{green}1,\color{blue}0,\color{blue}0,\color{blue}0,\color{green}1,\color{blue}0,\color{blue}0,\color{blue}0,\color{green}1,\underline{\color{blue}0,\color{blue}0,\color{green}1,\color{blue}0,\color{blue}0},$$ $$\underline{\color{blue}0,\color{green}1,\color{blue}0,\color{blue}0,\color{blue}0,\color{green}1,\color{blue}0,\color{blue}0,\color{green}1,\color{blue}0,\color{blue}0},\color{blue}0,\color{green}1,\color{blue}0,\color{blue}0,\color{green}1,\color{blue}0,\color{blue}0,\color{blue}0,\color{green}1,\color{blue}0.$$
If we submit this sequence (42 terms, the underlined ones belong to the previous sequence of $16$) to the OEIS, we find that they match with the entry A221150, i.e. with the terms of a generalized Fibonacci word, meaning that there could be a strong relation between this Penrose tiling and the Fibonacci word fractal, illustrated below.
(The intuition would suggest me that the empty, squared substructures of the fractal should correspond to the structures with central pentagonal symmetry in the tiling).
My conjecture is that the match between the binary sequence obtained from the Penrose tiling and the one related to the Fibonacci word fractal must keep independently on the (even) number of generations.
I believe that, if the conjecture is true, the reason should be related to the Golden ratio (which is at the base of both these mathematical objects). However, I cannot provide a proof of such conjecture.
Therefore I thank you very much for any comment, suggestion or reference which might help me to explain such (beautiful) connection.
I apologize with the experts in case this is a trivial problem, and for incorrectness. Thanks again!