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A Multidimensional Continued Fraction Generalization of Stern’s Diatomic Sequence
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Thomas Garrity

Department of Mathematics and Statistics

Williams College

Williamstown, MA 01267

USA

**Abstract:**

Continued fractions are linked to Stern's diatomic sequence 0, 1, 1, 2,
1, 3, 2, 3, 1, 4, ... (given by the recursion relations
α_{2n} = α_{n} and
α_{2n+1} = α_{n} +
α_{n+1}, where α_{0} = 0
and α_{1} = 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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