How to find the number of different arrangements of n squares with conditions that firstly every arrangement should be connected and secondly if two squares intersect they should intersect on a whole edge(not part of edge or a vertex)? For example in the image below two arrangement on the left are allowed and the two on the right are not.
and for n=4 there are five different arrangements as shown below.
How one should study this problem in an abstract mathematical way? Is there some kind of a partition of a number to be related to this?
edit 1: symmetries (rotations and reflections) are not allowed.