2006/10/19 by Bruce Reznick, Reznick, Bruce · 3 citations
Computer Science · Mathematics · #05A15 #11B37 #11B57 #11B75 #Advanced Algebra and Logic #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.math/0610601
openalex publication_date 2006/10/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.