A Quad-Omino (not to be confused with a tetromino) is a square-shaped tile with numbers in each corner. Like the game of dominoes, the object is to place them next to each other so their numbers match.

The standard set I own with numbers 0 to 5 has 125 tiles. Does the set really contain every possible tile?

Find a formula for the total number of different tetrominoes in a set numbered from 0 to n.