Who discovered the following formula for finding the next consecutive square? I know I'm not the first to discover this while being bored in seventh-grade math "learning" about perfect squares. The formula I discovered is this:
Sorry guys, I tried expressing it algebraically but I'm not very good with numbers, so I'll do my best to tell you in English.


*

*Take the square of $0$ : $0$

*Then take the square of the next whole number: $1^2=0^2+1=1$

*Then take the square of the next whole number: $2^2=1^2+3=1$


As you can see there is a pattern here. The power of a whole number is the number of the odd number that you can add to the square of the las number. If this is confusing then this should help : $1=1$  $3=2$ $5=3$ ...
Can someone just tell me who first discovered this?
(Sorry for the lack of simple notation once again. In case it wasn't clear, my mathematical studies have not even taken me to the end of elementary prealgebra.)

I'm not sure who first discovered this, but notice this: $(n+1)^2=n^2+2n+1$.  The $2n+1$ is an odd number which is being added to the square of $n$ to obtain the square of $n+1$.  Just like you discovered yourself.
Using the method of induction, one may prove without any doubt that $n^2=1+3+5+\cdots+(2n-3)+(2n-1)$.
You might have heard of Babbage's difference engine.  Polynomials are used quite often to approximate more complicated functions, and people used to go through considerable effort using them to make books full of important numbers (for instance, logarithm tables).  One thing people noticed is that you can compute the values of a polynomial by doing repeated additions using the method of divided differences (if you want to see what these are: try taking a polynomial, say $n^2+n+1$, making a table of the polynomial for $n=1,2,3,4,\ldots$, then taking the difference between consecutive values, then the differences of the differences, then the differences of those differences again.  Notice a pattern?  You must have already done something like this for $n^2$.)  The difference engine used gears to then compute the values of polynomials much faster than anyone could do by hand: just as fast as someone could write down the numbers from the display!
As a bonus, here is a graphical proof that the sum of the first odd numbers is a square: http://proofsfromthebook.com/2013/12/05/sum-of-the-first-n-odd-integers-is-a-square/
