All Self-Avoiding Paths Through a 2-D Grid

There are 2 self-avoiding (non-self-intersecting) paths through a 1 × 1 grid:
2 paths, 1 x 1 grid

12 paths through a 2 × 2 grid:
12 paths, 2 x 2 grid

184 paths through a 3 × 3 grid:
184 paths, 3 x 3 grid

While we’re on the subject, there seem to be 18 self-avoiding paths through a 1 × 1 × 1 lattice:
18 paths, 1 x 1 x 1 lattice

156 paths through a 2 × 1 × 1 lattice:
156 paths, 2 x 1 x 1 lattice

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