You might know that the number of tilings of a $2 \times 1$ rectangle is $F_n$. For example, the number of tilings here of this $2 \times 10$ rectangle is $F_{10} = 89$:
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If we had a really big $2 \times n$ rectangle and $n \gg 1$ and tiled that, I could try to draw a $2 \times 4$ box and see what it covers. And we could have a tiling with some defects:
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or (this one does not look possible)
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or
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Have I missed any?
I am trying to imagine the possible shifts if we move our imaginary $2 \times 4$ rectangle a unit to the right (+1) or to the left (-1) we can end up with one of the boundary conditions above. So I'm trying to figure out the odds of getting each kind of pattern in box $2 \times 4 \subset 2 \times n$, but also the transition matrix between these patterns under the shift maps $T^{\pm1}: (x,y) \mapsto (x \pm 1, y)$.