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This is basically three intertwined sequences:
triangular: 3 x 1, 3 x 4, 3 x 16, ... 3 x 4 ^ n
square: 4 x 1, 4 x 9, 4 x 81, ... 4 x 9 ^ n
hexagonal: 6 x 1, 6 x 7, 6 x 49, ... 6 x 7 ^ n
Each number in the above sequence has a particular geometric interpretation:
3: a single triangular tile
4: a single square tile
6: a single hexagonal tile
12 = 3 x 4 = triangle (three sides) x 4 tiles = one triangle in the centre, with 3 equally-sized triangles surrounding it
36 = 4 x 9 = square (four sides) x 9 tiles = one square with 8 other similar copies surrounding it
42 = 6 x 7 = hexagon (six sides) x 7 tiles = one hexagon with 6 other copies surrounding it
Each number in the sequence has a prime factorisation which uniquely determines whether it corresponds to a triangular, square or hexagonal tiling and the tiling's size. Factorisation and rewriting into canonical form effectively becomes the inverse operation of "mixing" the three initial sequences.
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