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I remember that the recursion

$r(0)=0, \ \ r(n+1)=\frac{1}{2 [r(n)]+1-r(n)}$ produces a sequence of rational values $ 0 \mapsto 1 \mapsto 1/2 \mapsto 2 \mapsto 1/3 \mapsto ... $ which exausts the positive fractions (and of course every fraction can only appear once).

Unfortunately I do not remember the reference for this statement. Can anybody help me?

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Have a look at chapter 19 ("Sets, Functions and the continuum hypothesis") in "Proofs from THE BOOK" (5th edition) by Aigner/ Ziegler.

The sequence originates in the paper "Recounting the Rationals" by Calkin/Wilf but in this paper you can't find the formula you mention.

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  • $\begingroup$ Thank you! Actually the keyword "Calkin/Wilf" is really useful: with this you can find also several references online (Wikipedia, OEIS, etc.). en.wikipedia.org/wiki/… $\endgroup$ – ccarminat Mar 14 '15 at 18:06
  • $\begingroup$ The answers to mathoverflow.net/questions/200656/… also contain a lot information. $\endgroup$ – gsa Mar 23 '15 at 15:30

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