How to explain the patterns in the matrix defined in François Brunault's answer to the question Freeness of a Z[x] module depicted below? --

Choosing colors according to the highest power of 2 which divides an entry, we can obtain for example the following pictures:

Taking instead highest power of 3 dividing an entry, we can obtain for example this picture:

Coloring zero entries white, negative entries red and positive entries blue yields the following picture:

And finally, marking non-zero entries black and leaving zero entries blank, the pattern looks like this:

**Added:** Assigning colors according to numbers of prime factors of the entries,
we can obtain the following picture:

**Added on June 15:** Choosing colors depending on entries (mod 47), we can obtain
the following picture (nonzero entries divisible by 47 are colored red):