Here is part of a tiling, from https://cs.uwaterloo.ca/~csk/hat/app.html - 4-th level at South West region. I colored obvious trivial patterns of some dark blue hats: red, green and orange. Can those be proved by induction after suitable formalization of the substitution rules? The interesting pattern, in purple and orange, starts with this sequence of orange hats: 1, 4, 12. What is the general term of this sequence and is it familiar?
EDIT: A preliminary analysis of some recurrences suggests that the sequence of positions of purple hats are Fibonacci numbers: they are $\phi_{2n} \ \ (1,3,8,21,55,...)$. A proof of this is needed.