Reznick, Bruce Regularity properties of the Stern enumeration of the rationals. (English) Zbl 1204.11027 J. Integer Seq. 11, No. 4, Article ID 08.4.1, 17 p. (2008). Summary: 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 \(s(n)/ s(n+1)\). We show that in a strong sense, the average value of these fractions is \(3/2\). We also show that for \(d\geq 2\), 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 \(3\). Cited in 2 ReviewsCited in 12 Documents MSC: 11B37 Recurrences 11B83 Special sequences and polynomials Keywords:Stern sequence; enumerations of the rationals; Stern-Brocot array; Dijkstra’s “fusc” sequence; integer sequences mod m Software:OEIS PDFBibTeX XMLCite \textit{B. Reznick}, J. Integer Seq. 11, No. 4, Article ID 08.4.1, 17 p. (2008; Zbl 1204.11027) Full Text: arXiv EuDML EMIS Online Encyclopedia of Integer Sequences: Stern’s diatomic series (or Stern-Brocot sequence): a(0) = 0, a(1) = 1; for n > 0: a(2*n) = a(n), a(2*n+1) = a(n) + a(n+1). Numbers m such that Stern’s diatomic A002487(m) is divisible by 3. a(0)=a(1)=0, a(2)=a(3)=2; for n >= 3, a(n) = a(n-1) + 4*a(n-3).