Recamán Sequence

0, 1, 3, 6, 2, 7, 13, 20, 12, 21, ...


Recamán Sequence is an integer sequence. It's particularly beautiful and interesting in terms of strange balance between order and chaos. There are two ways of discovernig the balance. The first one is to try to visualize it and the second one is to try to create an audio version of the sequence.


So, how to visualize it? 🤔

Well, let's say we have a counter. It's going to start at 1 and then go up (1, 2, 3, ...). For every counter value we're going to try to go backwards by counter value steps. If the spot is taken or there are no numbers backwards, we go forward.

So, we start at 0 with counter value = 1. We're trying to go backwards, and... Oh, no!!! There are no numbers in the sequence :( But we don't give up and go to 1.

Now we're trying to go backwards by 2. Unfortunately, we can't. So, we go forward by 2 and land on 3.

Now we want to go backwards by 3 and land on 1. But the spot is taken :^( So, we have no alternative but to go to 6.

Far from losing hope, we should face the challenges and try to go backwards by 4. And yay! The spot is open, so we land on 2.

Now we try to go backwards by 5. Well, we can't... But don't worry, let's go to 7!


And so on...

And finally let's visualize the Recamán Sequence and check out audio interpretation of it! As you'll see, it sounds kind of music that was made in the 1970s by Kraftwerk and other similar bands. It seems to me, that artists of that period would like the pattern that is drawn by the computer, too.

sound OFF sound ON

This visualization was made using tutorials by Daniel Shiffman and inspired by Numberphile's video about the sequence.