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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