Journal of Integer Sequences, Vol. 16 (2013), Article 13.7.7

A Multidimensional Continued Fraction Generalization of Stern’s Diatomic Sequence

Thomas Garrity
Department of Mathematics and Statistics
Williams College
Williamstown, MA 01267


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.

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(Concerned with sequences A002487, A228925.)

Received February 10 2013; revised version received February 21 2013; September 5 2013; September 8 2013. Published in Journal of Integer Sequences, September 8 2013.

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