Consider the sequence 1,1,2,1,3,2,3,1,4... where the ith element (starting from 0) represents the number of ways to write i as a sum of powers of 2, with each power appearing at most twice. Then you can enumerate the positive rationals without duplicates as follows: Q_n = S_n / S_(n+1). So Q_1 = 1, Q_2 = 1/2, Q_3 = 2/1, Q_4 = 1/3, etc. You can get all the rationals by inserting each one's negation after it. Q_1 = 1, Q_2 = -1, Q_3 = 1/2, Q_4 = -1/2, etc. See http://en.wikipedia.org/wiki/Calkin%E2%80%93Wilf_tree
