A Multidimensional Continued Fraction Generalization of Stern’s Diatomic Sequence

Continued fractions are linked to Stern’s diatomic sequence 0, 1, 1, 2, 1, 3, 2, 3, 1, 4, . . . (given by the recursion relations α2n = αn and α2n+1 = αn + αn+1, where α0 = 0 and α1 = 1), which has long been known. Using a particular multidimensional continued fraction algorithm (the Farey algorithm), we generalize the diatomic sequence to a sequence of numbers that quite naturally can be termed Stern’s triatomic sequence (or a two-dimensional Pascal sequence with memory). As both continued fractions and the diatomic sequence can be thought of as coming from a systematic subdivision of the unit interval, this new triatomic sequence arises by a systematic subdivision of a triangle. We discuss some of the algebraic properties of the triatomic sequence. 1 The classical Stern diatomic sequence The goal of this paper is to take a particular generalization of continued fractions (the Farey algorithm) and find the corresponding generalization of Stern’s diatomic sequence, which we will call Stern’s triatomic sequence. In this first section, though, we will quickly review the basics of Stern’s diatomic sequence (sequence A002487 in Sloane’s Encyclopedia). Hence experts should skip to the next section. In Section 2 we give a formal presentation of Stern’s triatomic sequence. Section 3 shows how Stern’s triatomic sequence can be derived from a systematic subdivision of a triangle. Section 4 shows how Stern’s triatomic sequence arises from the multidimensional continued fraction stemming from the Farey map, in analog to