I did this exact same thing while in high school, and by an interesting coincidence, I was doing it again last night, and was going to make a post on it! Although I do it all on paper, since I have yet to learn programming skills. At the time, I wrote what my geometry teacher called "telescoping equations" for each of the degrees. So for degree 3 it's already an unwieldy

((x^3 - (x-1)^3) - ((x-1)^3 - (x-2)^3)) - (((x-1)^3 - (x-2)^3) - ((x-2)^3 - (x-3)^3)) = 6

Pretty useless after I found out the factorial rule, like you did. :)

I tried a number of methods for the prime sequence. I tried to imagine them as absolute value differences for some other number sequence, and then look for some pattern in that. For example, if you start with 1 and then add 2, and then 3, and 5, and so on, the sequence eventually starts reproducing other prime numbers. A difference table of another sequence, where x is the nth position of these primes in the prime number sequence, produces a sequence that starts with

2, 5, 10, 17, 26, 41, 58

If we start with 0, a difference table on this sequence *almost* yields the start of prime number sequence.

```
0
2
2
3
5
5
10
7
17
9
26
15
41
17
58
```

If 26 were 28, it would be

```
0
2
2
3
5
5
10
7
17
11
28
13
41
17
58
```

It wouldn't matter anyway, because you'd have to start with the prime number sequence in order to get the process going. Still, I thought it might be interesting to see if I could find some sort of self-similarity.

I tried to alternate signs when doing these "backwards difference tables," then taking differences of their absolute values, but that achieved no interesting result, yielding a similar randomish growth pattern of "zero triangles."

I've tried to use Excel for this, but it's not as neat. I'm interested in learning how to write some programs. Would you be willing to share some of your source code with me, Job? I am a complete beginner and don't know where to start.