Geodesic
Regularity properties of the Stern enumeration of the rationals
Journal of integer sequences, Tome 11 (2008) no. 4
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 $n \ge 1$. We show that in a strong sense, the average value of these fractions is 3/2. We also show that for $d \ge 2$, 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.
Classification : 05A15, 11B37, 11B57, 11B75
Keywords: stern sequence, enumerations of the rationals, stern-brocot array, dijkstra's "fusc" sequence, integer sequences mod m 
@article{JIS_2008__11_4_a5,
     author = {Reznick,  Bruce},
     title = {Regularity properties of the {Stern} enumeration of the rationals},
     journal = {Journal of integer sequences},
     year = {2008},
     volume = {11},
     number = {4},
     zbl = {1204.11027},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2008__11_4_a5/}
}
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Reznick,  Bruce. Regularity properties of the Stern enumeration of the rationals. Journal of integer sequences, Tome 11 (2008) no. 4. http://geodesic.mathdoc.fr/item/JIS_2008__11_4_a5/