Different shapes made from particular number of squares

combinatorics

Solution 1:

When you have such a list of integers, it's always worthwhile to search for it in the Online Encyclopedia of Integer Sequences. The two sequences you describe are OEIS sequence A000105 and OEIS sequence A001168, though it seems you got the last counts wrong and they should be $369$ and $216$, respectively.

See also MathWorld and Wikipedia on polyominoes. According to the MathWorld article, not even the growth rate is known, so no formula is known for these numbers. As to your question why this is so, that's rather hard to answer specifically. Many mathematical problems prove intractable.

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