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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 $n=4$:

enter image description here

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.

Shaun
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    ah. "What is the next term in 1,2,3,..." 4, of course! nope. 13! – Guy Mar 13 '14 at 11:24
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    Configs 7 and 11 are equivalent, it seems. – ccorn Mar 13 '14 at 11:27
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    I just noticed: I've corrected it. Thank you :) – Shaun Mar 13 '14 at 11:27
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    Analytic Combinatorics may be of help. Try constructing a class of combinatorial objects you require and then apply a transfer function to obtain a generating function which you can solve explicitly or asymptotically. – pshmath0 Mar 13 '14 at 11:35
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    Just maybe a representation as trees might lead somewhere, similar as in http://math.stackexchange.com/questions/651317/catalan-numbers-staircase-bijection ? – Wouter M. Mar 13 '14 at 11:37
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    OEIS is generally kept quite up to date. What's known about the sequence is probably what's there. – Gerry Myerson Mar 13 '14 at 12:01
  • @GerryMyerson: In that case I'm (surprised and) very sorry. Thank you. [This didn't seem up to MO standards so . . . ] – Shaun Mar 13 '14 at 12:05
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    There is no good reason there should be a reasonable analysis of this count. Maybe there is a bit better chance to get the count where you do not rule out symmetric repeats. – GEdgar Mar 13 '14 at 12:54

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