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Regularity properties of the Stern enumeration of the rationals. (English) Zbl 1204.11027

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 \(3\).

MSC:

11B37 Recurrences
11B83 Special sequences and polynomials

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