You might be wondering what in the world that is and why anyone would care. Well, bro, it turns out that Gould’s sequence is a pretty cool integer sequence named after some dude named Henry W. Gould (whoever he was).

So, let’s break this down real quick an “integer sequence” just means a list of numbers where each number is calculated based on the previous one. In this case, we’re talking about a specific type of integer sequence called Pascal’s triangle. If you’ve ever seen that thing with all those little boxes and numbers inside them, well, that’s Pascal’s triangle!

Now, here’s where things get interesting Gould’s sequence is actually just counting the number of odd integers in each row of Pascal’s triangle. That might sound boring at first, but trust us when we say it gets pretty wild once you start looking into it.

For example, let’s take a look at the first few numbers in Gould’s sequence: 1, 2, 2, 4, 2, 4, 4, 8… Do you see what’s happening here? The number of odd integers is doubling with each row! That means that if we keep going down the list, we’re eventually going to hit some pretty big numbers.

But wait there’s more! Gould’s sequence isn’t just a bunch of random numbers. It actually has this really cool self-similar sawtooth shape (check out the image below). And get this if you take the partial sums of Gould’s sequence, they grow proportionally to n log 2… but with a constant of proportionality that oscillates between 0.812556 and 1!

It has some pretty wild properties (like doubling with each row) and a self-similar sawtooth shape. And if you take its partial sums, they grow proportionally to n log 2… but with this oscillating constant!

Who needs math textbooks when you can learn about Gould’s sequence at the local dive bar? Cheers!