What are the next several lines and how are they formed?

Do the sequences have any special feature?

Does the pattern continue indefinitely?

This certainly seems to work.  I assume by the wording of Tristan's explanation, that the fraction sequence should actually look like:

0/1, 1/6, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 5/6, 1/1

(lowest to highest in numerical value)

The fact that the sequence is therefore palindromic is not by construction, but falls out of the math.