Regularity properties of the Stern enumeration of the rationals

TL;DR

This paper investigates the regularity and distribution properties of the Stern sequence, revealing average ratios and uniform distribution of pairs modulo various integers, with detailed results for mod 2 and 3.

Contribution

It provides new insights into the distribution and average behavior of the Stern sequence, including uniformity in residue classes and precise modular results.

Findings

Average value of s(n)/s(n+1) is 3/2.

Pairs (s(n), s(n+1)) are uniformly distributed modulo d.

Detailed modular distribution results for d=2 and 3.

Abstract

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 for every pair of relatively prime positive integers (a,b), there exists a unique n so that s(n) = a and s(n+1) = b. We show that, in a strong sense, the average value of s(n)/s(n+1) is 3/2, and that for all d, (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.