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A few days prior to the current moment, I was browsing through some Numberphile videos, upon coming across a video about the Recaman sequence. The Recaman sequence is defined as being:

if n > 0, and the term is not already included in the sequence, a(n) = a(n-1)-n else a(n) = a(n-1)+n

Having an insatiable curiosity, I decided to investigate the quotients of each group of two successive terms in the Recaman sequence. The experimenting resulted as such:

1 , 3 , 6 , 2 , 7 , 13 , 20 , 12;

3 , 2 , 1/3 , 7/2 , 13/7 , 20/13 , 12/20

As you may be conscious of from simply skimming over this experimenting, I ultimately found the quotients of every two consecutive terms. Of course, I repeated this process until I eventually came to a final number, which happened to equate to 2.03329e-12. Take heed of the fact that there is a -12 present, and recall that I was experimenting with Recaman numbers up to 12. Why is the last term in the provided Recaman sequence equal two the absolute value of the eventual quotient of the consecutive terms? I found myself pondering whether there was any mathematical reason for such, bringing me here.

This question led to myself to experiment with other numbers, yet I could not seem to find anything, where the last term in the provided Recaman sequence was equal to the absolute value of the "eventual" quotient of the consecutive terms. I attempted to postulate a reason for such, listing out some of the mathematical properties of the number 12:

  • abundant number
  • pentagonal number
  • precedes a prime number and succeeds a prime number
  • product of the first three factorials

How would this relate to the Recaman sequence, or is my train of thought simply pointed in an erroneous direction?

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