Overlapping Squares

The above consists of $$n$$ number of $$2\times2$$ squares such that each adjacent $$2\times2$$ squares overlap each other on exactly one $$1\times1$$ square. And I can only move 1 unit down or 1 unit to the right at a time.

Let $$M_n$$ denote the total number paths for me to choose from such that I start the top left point on the top, X til I to move to the bottom right point on the bottom, Y.

Find $$\displaystyle \lim_{n\to\infty} \dfrac{M_{n+1}}{M_n}$$.

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