Start with n₀ = 0

Now, you’ll make steps of increasing size: 1, 2, 3...

First, you’ll try to go back from where you are (so from 0 you’d try to go to  − 1) but you must obey two rules:

1. The new “landing site” must be greater than 0, and
2. The new “landing site” must be new in the sequence (it has never appeared before)

If you can’t go backwards, you go forwards.

So from 0, the next step is of size 1. If you try to go backwards, you’ll land at -1. This is against the rules, so you go forward to 1.

Now from 1, the next step is of size 2. If you try to go back, you’ll land at -1. This is against the rules, so you go forward to 3.

From 3 with a step size of 3, you go to 6.

From 6 with a step size of 4 you go to 2 (because it’s greater than 0 and has never appeared before)

Named after Bernardo Recamán, but not by Recamán himself.

It is conjectured that all natural numbers appear in this sequence, but has yet to be proven. Drawing this sequence as a series of semi-circles drawn along a number line produces pretty pictures.

References and Bibliography

• OEIS Foundation Inc. (Ed.). (2018, September 30). Recamán’s sequence. OEIS Foundation Inc. https://oeis.org/w/index.php?title=Recamán's_sequence&oldid=1625628
• Weisstein, E. W. (2003, October 27). Recamán’s sequence: From MathWorld–a wolfram web resource. (MathWorld, Ed.). https://mathworld.wolfram.com/RecamansSequence.html
• Wikipedia contributors. (2021). Recamán’s sequence — Wikipedia, the free encyclopedia. https://en.wikipedia.org/w/index.php?title=Recam%C3%A1n%27s_sequence&oldid=1019874201.