So this is a bit of math that is simple to understand (you could teach it to a bunch of sixth graders without difficulty), was completely undiscovered until the 1950s, but feels like one of those things that should be as old as the Pythagorean Theorem. It's called "Moessner's Theorem".
Here's a sample:
- Take the natural numbers: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 ...
- Now cross off every third number: 1 2
34 567 8910 111213 141516 ... - Take the partial sums of what's left: 1 3 7 12 19 27 37 48 61 75 91 ...
- Now cross every second number: 1
371219273748617591 ... - Take the partial sums of what's left: 1 8 27 64 125 216 ...
Suddenly, you have the cubes of the natural numbers!
That's the pattern: pick a starting gap size (I used three above, but pick any integer > 1), then start with the natural numbers, cross off every {{GAP SIZE}} values and take the partial sums of what's left. Then repeat with {{GAP SIZE}} - 1, etc, until you have crossed off every second number and taken the partial sums there. What you're left with is the natural numbers to whatever power your initial {{GAP SIZE}} was.
If you take a starting {{GAP SIZE}} of 2, you just get that the partial sums of the odd natural numbers is the squares; that case was known in ancient times and is pretty easy to prove in a bunch of ways. But this way of generalizing that wasn't ever published until 1951.
The person who discovered this (Moessner) couldn't prove it, though it was proven within a year of him first publishing this trick. There's also variations on this alternating "take away some things and then make the partial sums" pattern, like this where what I'm doing in each line where I take things away is taking away the triangular-numbered elements (i.e. the first, third, sixth, tenth, fifteenth, etc):
Note how the first values I take away each time are the factorial numbers: 1, 2, 6, 24, 120, ...
Anyway, I haven't proved any of this myself nor have I really tried; I just ran across a mention of this this morning and haven't really had time to play with it but it's kind of cute, isn't it?