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Consider the number of leaf-free subgraphs of the $2 \times n$ grid—which is to say, the number of ways to draw lines on the $2 \times n$ grid such that no grid point has degree exactly 1.

For example, when $n = 4$ there are $15$ such subgraphs:

The number of leaf-free subgraphs of the $2 \times n$ grid is given by the linear recurrence

$a(1) = 1$ $a(2) = 2$ $a(n+1) = 5\left(a(n) - a(n-1)\right)$.

What is a combinatorial explanation for this recurrence?