Journal of Integer Sequences, Vol. 11 (2008), Article 08.4.1

Regularity Properties of the Stern Enumeration of the Rationals

Bruce Reznick
Department of Mathematics
University of Illinois
Urbana, IL 61801


The tern 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.

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(Concerned with sequence A002487.)

Received August 31 2008; revised version received September 16 2008. Published in Journal of Integer Sequences, September 16 2008.

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