The Challenge: Create a visual (or audio?) of the Recamán sequence, created by a Colombian mathematician, Bernardo Recamán Santos (who seems to have very little biographical information out there??). I was first introduced through Alex Bellos and Edmund Harris’s book.
Materials Needed: Straight edge, maybe a compass, maybe a ruler. Paper, riting utensil.
Math concepts you could explore with this challenge: algebra, circles, counting, functions, proportions/ratios, randomness, sequences
The formal rule for the sequence is:
In other words, start at zero, and then add 1, then add 2, then add 3 and continue using the next integer to get the next number BUT if you can go back, go back. So after you’ve added 3, you’ll be at 6, and will then be able to go back to 2. This video explains it all much better than myself. I also love that the video explores the sequence in audio form. I’ve been challenged recently to think of how we might think of ways that math is exclusionary, and I think that insistence that it’s written down is a constraint we could lift. I love that this explores it’s audio manifestation.
I also love that from relatively simple rules, we get such complexity. If you watch my video up above, you can maybe pick out the spot where I had to stop counting and needed to work the sequence out on paper, which nearly led me to asking whether or not the Online Encyclopedia of Integer Sequences was incorrect as 42 appears twice, but sure enough, it is in the sequence twice, once reach from above and once from below. Which makes me wonder what the sequences of numbers is that’s doubled in this sequence? Maybe tripled? Is that even possible? I wonder. Infinity offers a lot of room for exploration.
Reflection questions:
- How else might we represent this sequence?
- What sequences result from tweaking some of the rules for this sequence?
Depending on how you use this activity, you may engage with different mathematical standards. I’ve listed possible connected math content above. Here are a few suggestions for how you might integrate the 8 mathematical practices. Feel free to add your own suggestions in the comments!
7.) Look for and make use of structure. How does the audio version help us learn more about the sequence? Is there something you can learn from the visual or audio version that gives you more or less insight into the sequence?
8.) Look for and express regularity in repeated reasoning. What is always true about the sequence values? What could never be true?
arbitrarilyclose 
Interesting fact: the number 852655 doesn’t appear in the Recamán sequence at least in the first 10^73 elements (that’s ten undecillions ish). There’s no proof showing it doesn’t come up at some later stage though. Or that it does.
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That’s fascinating! Thanks for sharing, I wondered if this was the (possible) case!
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Actually, my bad. It’s 10^15, not 10^73. Still interesting though.
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