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?



Enumerating positive fractions (reference missing)
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.


