Regularity properties of the stern enumeration of the rationals

The Stern sequence s(n) is defined by s(0) = 0,s(1) = 1, s(2n) = s(n), s(2n+ 1) = s(n) + s(n + 1). Stern showed in 1858 that gcd(s(n),s(n + 1)) = 1, and that every positive rational number a/b occurs exactly once in the form s(n)/s(n+1) for some n ≥ 1. We show that in a strong sense, the average value of these fractions is 3/2. We also show that for d ≥ 2, the pair (s(n), s(n+ 1)) is uniformly distributed among all feasible pairs of congruence classes modulo d. More precise results are presented for d = 2 and 3.