A formula for the number of structures composed of n squares

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Main Question or Discussion Point

Hello,
The problem I came up with deals with the structures that can be obtained by joining squares side to side or corner to corner. Specifically to this problem, structures, that are symmetrical to each other, are regarded the same.
Ideally, I am looking for a formula that will tell how many structures can be made with n given squares. However, I am aware of the problem's complexity (this is the nature of partition problems...), so,at least, it will be comforting, if someone comes up with an insight about what happens "behind the scenes" in the problem.
One can easily observe that the n+1 structures can be generated from the n structures. By applying this principle, I was able to draw all the unique structures up to n=5. Then, I tried to classify them by the minimal rectangle that encloses the structure, and still lead to no progress.
I ran out of new strategies for a solution, and I am asking for another perspective on the problem.

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jedishrfu
Mentor
Try thinking of it as a sequence given $a_n$ what is $a_{n+1}$ ?

mfb
Mentor
You have 5 numbers - did you search for the beginning of the sequence on OEIS?

These problems often end up in a lot of casework done by a computer.
Try thinking of it as a sequence given $a_n$ what is $a_{n+1}$ ?
I would be very surprised if you get a nice formula for that.

Wolfram mathworld calls it a polyplet and Wikipedia a pseudo-polyomino or polyking. The OEIS also has various sequences, depending on wether rotations and/or reflections will result in the same figure, and what to do with holes. No one has any formula.
There's much more literature on poliyominoes, that only allow edge-connections.

mfb