<s>[INST] What is a simple formula that can create the patterns $+1 +2 +3 +4$, or $+1 +3 +5 +7$, or $+1 +4 + 7 +10$, etc. Originally I was looking for some simpler more intuitive way to appreciate the value of squaring something. I looked for how much it was increasing by every time I add $+1$ to the size of what was being squared. so in other words $3$ squared is $+5$ more than $2$ squared, and $4$ squared is $+7$ more than $3$ squared, and $5$ squared is $9$ more than $4$ squared. You can see that whenever you increase the number being squared by $1$, the amount increased itself increases by $+2$ more than the previous increase. ie, the increase itself increases by a rate of $+ 2$.
So
$$- 1^2 - 0^2 = +1$$
$$- 2^2 - 1^2 = +3$$
$$- 3^2 - 2^2 = +5$$
$$- 4^2 - 3^2 = +7$$
$$- 5^2 - 4^2 = +9$$
and

*

*the difference between the $+3$ and $+1$ is $2$

*the difference between the $+5$ and $+3$ is $2$

*the difference between the $+7$ and $+5$ is $2$

*the difference between the $+9$ and $+7$ is $2$  etc...

So I say, it's "increase bonus" is at a rate of +2. That's definitely not a correct mathematical term, but in my ignorance that's what i'm calling it.
This struck me as very interesting, because I used to make little wc3 mods as a kid and they gave tools that made same pattern. What I became interested in, was what if I wanted it to increase by a pattern of say +1, +2, +3, +4, +5. (for an increase bonus of +1). Or what if I wanted it to be +1, +4, +7, +10, +13 (an increase bonus of +3)?
My main question is, is there a formula where I can easily change the "increase bonus" to whatever rate I want? While I can do this manually, I don't know how to represent it as a simple, calculable formula.
My side question is... what is this formula / branch of mathematics actually officially called?
 [/INST] A polynomial of degree $n$ will have the $n$th finite difference equal to $n!$ times the leading coefficient of the polynomial.  In your example, you have
\begin{array}{|c|c|c|c|}
\hline
x & x^2 & \text{1st difference} & \text{2nd difference} \\
\hline
1 & 1 & & \\
2 & 4 & 3 & \\
3 & 9 & 5 & 2 \\
4 & 16 & 7 & 2 \\
5 & 25 & 9 & 2 \\
\hline
\end{array}
If you want the bonus to be $+3$ instead, you can use $\frac32x^2$ instead of $x^2$:
\begin{array}{|c|c|c|c|}
\hline
x & \frac32x^2 & \text{1st difference} & \text{2nd difference} \\
\hline
1 & \frac32 & & \\
2 & 6 & \frac92 & \\
3 & \frac{27}2 & \frac{15}2 & 3 \\
4 & 24 & \frac{21}2 & 3 \\
5 & \frac{75}2 & \frac{27}2 & 3 \\
\hline
\end{array}
Or, if you want instead the third finite difference to be $+3$, you can use $\frac12x^3:$
\begin{array}{|c|c|c|c|c|}
\hline
x & \frac12x^3 & \text{1st difference} & \text{2nd difference} & \text{3rd difference}\\
\hline
1 & \frac12 & & & \\
2 & 4 & \frac72 & & \\
3 & \frac{27}2 & \frac{19}2 & 6 & \\
4 & 32 & \frac{37}2 & 9 & 3\\
5 & \frac{125}2 & \frac{61}2 & 12 & 3\\
\hline
\end{array}
</s>