I've been trying to describe mathematically the $n$th term $a_n$ of the sequence A224239. We get $a_n$ by counting the distinct ways to fill an $n\times n$ grid with squares of smaller integer size, up to the symmetries of the $n\times n$ square. It's much more difficult than I expected when I started playing around with it.
I believe these are the different configurations for $n=4$:
The OEIS page gives pictures for $n=5$.
How would you go about describing it mathematically? What's known about the sequence?
(You know what I mean when I say "describe mathematically", right?)
Thoughts: I think it's related to Waring's problem. The OEIS gives several related sequences that demonstrate that the one in question gets gigantic very quickly, so is difficult to experiment with (at least for me). I haven't come up with any explicit equations for $a_n$ I'm afraid.