Taylor The Roof Tiler

Taylor the roof tiler wants to know how many different perimeters can be obtained by tiling \(N\) tiles on a grid. The tiles are squares with side length 1 meter and are placed in such a way so that each tile is joined with at least one other tile.

There are 4 different ways of tiling \(N=2\) tiles as shown above. The sum of the possible distinct perimeters for \(N=2\) tiles is 8+6=14 (meters).

What is the sum of the possible distinct perimeters for \(N=7\) tiles?

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