Calculus How To

Recamán Sequence: Definition & Creating Terms

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What is a Recamán sequence?

There are two different sequences attributed to Columbian puzzle maker Bernardo Recamán Santos:

Recamán sequence #1 (entry A5132 in the OEIS) is usually the one referred to as “The” Recamán Sequence. It is defined by [1]:


  • an = an-1 – n, if an-1 – n > 0 and an-1 has already occurred in the sequence.
  • Otherwise, an = an-1 + n.

The first few numbers are:
{1, 3, 6, 2, 7, 13, 20, …}.

Recamán sequence #2 (entry A8336 in the OEIS) is defined by:

  • an+1 = an / n if n divides an

The first few numbers are:
{1, 1, 2, 6, 24, 120, …}.

Intuitive way to Create the Terms of the Recamán Sequence

Neil Sloane of ATT Labs (creator of OEIS) shed a little light on how the sequence is created in a 2008 Math Factor podcast [2].

Step 1: Write down the numbers
1 2 3 4 5 6 7 8 9
with gaps (we’re going to write the terms in those gaps).

Step 2: Write down the first term, which is 1, between the 1 and the 2:
1 (1) 2 3 4 5 6 7 8 9.

Step 2: Calculate the second term.
Ask yourself the question: Can we subtract 2 (the second number in our list) from 1 (the first term)?. We can’t (negative numbers or zero are not allowed), so we have to add 2 instead. 2 + 1 = 3, so:
1 (1) 2 (3) 3 4 5 6 7 8 9.

Step 2: Calculate the third term in the same way.
Ask yourself the question: Can we subtract 3 (the third number in our list from 3 (the second term). We can’t, so we have to add: 3 + 3 = 6, so:
1 (1) 2 (3) 3 (6) 4 5 6 7 8 9.

Continue in this way, obeying the basic rules:
Numbers can’t be zero, negative, or something already in the sequence.
If any of these conditions appear, just add instead of subtract. Note that if you add and get a number that has appeared before, that’s OK. It’s only not allowed if you subtract.

Deciphering Recaman’s sequence

Although 10230 terms of Recaman’s sequence have been computed, but it still remains a mystery [2] despite the OEIS’s calculation of an enormous number of terms. No one knows if every number will eventually appear; Those dedicated to deciphering the sequence have dubbed it “How to Recamán’s life” [3].

As of the January 2018, Prime Curious! reported that the smallest missing prime in the Sequence is 966727 (as of January 2018).

References

[1] Ding, C. (2012). Sequences and Their Applications. Springer London.
[2] Math Factor Podcast. Retrieved April 9, 2021 from: http://mathfactor.uark.edu/2008/05/dw-the-online-encyclopedia-of-integer-sequences/
[2] Myers, J. et al. (2020). Three Cousins of Recaman’s Sequence. Retrieved April 9, 2021 from: https://ui.adsabs.harvard.edu/abs/2020arXiv200414000M/abstract
[3] Roberts, S. (2015). Genius At Play: The Curious Mind of John Horton Conway. Bloomsbury Publishing.

CITE THIS AS:
Stephanie Glen. "Recamán Sequence: Definition & Creating Terms" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/recaman-sequence-definition/
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