Different shapes made from particular number of squares Good day! I’m currently investigating how different shapes can be made from a particular number of squares.
I have two major concerns: (1) Will there be a formula predicting the number of shapes that can be made from a certain number of squares without the flipping and rotation of formed shapes; and (2) how about with flipping and rotation?
Researching and manually listing the possible shapes that can be formed, my observations are the following:

Without flip and rotation:

Number of squares   -Number of shapes made

1- 1

2- 1

3- 2

4- 5

5- 12

6- 35

7- 108

8- 384 edit: should be 369

With flip and rotation:

Number of squares- Number of shapes made

1- 1

2- 2

3- 6

4- 19

5- 63

6- 208 edit: should be 216

Any help you might extend will be highly appreciated. 

PS. Will really, really be grateful if you can provide an explanation why there is (or there is no) formula that can be derived from this. 

Many thanks!

Edit: Added a picture because I don't think I was able to point my message clearly :D
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.
I thought of this question from playing blokus, and I thought I came up with an equation that works, but it only works for 1-5 pieces. I thought it was right since blokus doesn’t have pieces any larger than that. Here it is anyway though! For y= the size and x= number of pieces, $y=2^{(x-1)}-x+1$