Show that $T(n) = 4 × T (n - 1) - T(n-2)$...

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$T(n)$ is the number of spanning trees for the n-ladder.

Show that $T(n) = 4 × T (n - 1) - T(n-2)$.

-My teachers hint was to first show $T(n) = 3×T(n-1) + 2×T(n-2) + 2 × T(n-3) + ... + 2× T(2) + 3 × T(1) $

-I was unsure of what to do so I drew out the case for a 1-ladder, 2-ladder and 3-ladder. enter image description here

-Even with these initial cases done, I am really unsure of where to go with this problem, any help is appreciated.