I've been experimenting with combinations. Specifically, I've found some patterns:
The sequence $n \choose 2$ has a constant second difference and the sequence $n \choose 3$ has a constant third difference. I imagine this pattern continues.
Specifically, I was able to derive the sequence $a_{n+1} = a_n + n$ for $n \choose 2$ and have not derived one for $n \choose 3$ (I'm not sure how to approach this one).
What mathematical property gives way to what I am witnessing? I had not thought combinations could be calculated without multiplication or factorials.
This phenomena that you are experiencing is simply due to Pascal's triangle. The entry of the $n^{\mathrm{th}}$ row and $k^{\mathrm{th}}$ column is given by $\begin{pmatrix} n \\ k \end{pmatrix}$. You are finding a pattern along some diagonal of this triangle!