Analogs of the Stern Sequence

Abstract

We present two infinite families of sequences that are analogous to the Stern sequence. Sequences in the first family enumerate the set of positive rational numbers, while sequences in the second family enumerate the set of positive rational numbers with either an even numerator or an even denominator.

Two trees enumerating the positive rationals

“…They can be considered as generalizations of the Calkin-Wilf tree, in the sense that they lead to sequences which enumerate the postive rationals and satisfy relations similar to (1), (2) and (3). However, these generalizations are quite different from those proposed by T. Mansour and M. Shattuck ([MS11] and [MS15]), by B. Bates and T. Mansour [BM11] and by S. H. Chan [Cha11]. Finally, we show that the sequences (t n ) and (s n ) are, together with the Calkin-Wilf sequence, the only sequences (u n ) which enumerate the positive rationals and are defined by u 0 = 0 and a recurrence relation of the form :…”

Enumerating Trees

“…Similar to the interpretation of the denominators and numerators of the Calkin-Wilf sequence as a combinatorial function, there are further combinatorial interpretations of these forests in [7].…”

“…Some interesting injective families over Q, all of whose members are Möbius transformations, have been found by S.H. Chan, see [7]. These give rather forests with a finite number of components, instead of isolated trees.…”

Forests of complex numbers

“…Every positive rational number occurs exactly once as a vertex in the Calkin-Wilf tree, and the geometry of the tree encodes beautiful arithmetical relations between rational numbers (Bates, Bunder, and Tognetti [1], Bates and Mansour [2], Calkin and Wilf [4], Chan [5], Dilcher and Stolarsky [6], Gibbons, Lester, and Bird [7], Han, Masuda, Singh, and Thiel [8], Mallows [10], Mansour and Shattuck [11], Nathanson [12,14,13], Reznick [15]). Let For example, if…”