A Multidimensional Continued Fraction Generalization of Stern’s Diatomic Sequence
Department of Mathematics and Statistics
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.
Full version: pdf,
(Concerned with sequences
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.
Journal of Integer Sequences home page