Math - Horizontal-vertical connection between congruent squares without repeated transformation,?

... solve combination.Im in 8th grade so either I discovered something new or I rediscovered something.This is a very abstract math question that I d

... solve combination.
I'm in 8th grade so either I discovered something new or I rediscovered something.
This is a very abstract math question that I doubt many can understand so I have to give examples at the bottom. The point is to start a conversation so please, if you have something to add please do so.
It looks very complex but I'm just putting a bunch of parameters.
Here's the question:
Your job is to make shapes, but you must follow 4 criteria.
1. You may only build shapes from congruent squares, and they can not overlap. Nothing else should be used.
2. All squares must be connected horizontally or vertically and not diagonally or overlapping. Only 90 degree angles should exist.
3. You can only use n squares and must use all of them.
4. A transformation of a shape can be made with "transformation, rotation, reflection and dilation". When counting the amount of shapes ALL duplicate shapes that can be formed by any transformation must be eliminated and not counted towards the total.
And finally, the actual question.
For given number n, solve for the amount of shapes possible that can be made following those four criteria. Then, solve for the entire pattern with n=1, n=2.. etc
If some equation to solve this exists, then what is it? How can we apply this topic in real-life?
Examples:
I'm going to start us off at n=1.
We can only use one square so the first number in the sequence is 1.
n=2
We can form a square, then add another square in any of the four directions.
But, according to rule #4 they are all rotations of each other, so really we can eliminate and say only one shape can be made. Described as "two squares that share one side".
n=3
We can make a L, or a straight line of 3 squares.
n=4
I got 5: a T shape, a square, the straight line, a L/angle shape and what resembles a Z if you look at it horizontally.
n=5
This post is getting long so that's up to you to figure.

or:... solve combination.I'm in 8th grade so either I discovered something new or I rediscovered something.This is a very abstract math question that I doubt many can understand so I have to give examples at the bottom. The point is to start a conversation so please, if you have something to add please do so.It looks very complex but I'm just putting a bunch of parameters.Here's the question:Your job is to make shapes, but you must follow 4 criteria.1. You may only build shapes from congruent squares, and they can not overlap. Nothing else should be used.2. All squares must be connected horizontally or vertically and not diagonally or overlapping. Only 90 degree angles should exist.3. You can only use n squares and must use all of them.4. A transformation of a shape can be made with \"transformation, rotation, reflection and dilation\". When counting the amount of shapes ALL duplicate shapes that can be formed by any transformation must be eliminated and not counted towards the total.And finally, the actual question.For given number n, solve for the amount of shapes possible that can be made following those four criteria. Then, solve for the entire pattern with n=1, n=2.. etcIf some equation to solve this exists, then what is it? How can we apply this topic in real-life?Examples: I'm going to start us off at n=1.We can only use one square so the first number in the sequence is 1.n=2We can form a square, then add another square in any of the four directions.But, according to rule #4 they are all rotations of each other, so really we can eliminate and say only one shape can be made. Described as \"two squares that share one side\".n=3We can make a L, or a straight line of 3 squares.n=4I got 5: a T shape, a square, the straight line, a L/angle shape and what resembles a Z if you look at it horizontally.n=5This post is getting long so that's up to you to figure.

or:Sorry to burst your bubble bro, but it's called combinatorics. Mathematicians in history loved playing this sort of game. Was an optional course and the first lesson involved a similar problem to what you describe but with dominoes. Went out the door and picked another one.

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