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Stern Sequences for a Family of Multidimensional Continued Fractions: TRIP-Stern Sequences
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Ilya Amburg

Center for Applied Mathematics

Cornell University

Ithaca, NY 14853

USA

Krishna Dasaratha

Department of Mathematics

Stanford University

Stanford, CA 94305

USA

Laure Flapan

Department of Mathematics

University of California, Los Angeles

Los Angeles, CA 90095

USA

Thomas Garrity

Department of Mathematics and Statistics

Williams College

Williamstown, MA 01267

USA

Chansoo Lee

Department of Computer Science

University of Michigan, Ann Arbor

Ann Arbor, MI 48109

USA

Cornelia Mihaila

Department of Mathematics

University of Texas, Austin

Austin, TX 78712

USA

Nicholas Neumann-Chun

Department of Mathematics and Statistics

Williams College

Williamstown, MA 01267

USA

Sarah Peluse

Department of Mathematics

Stanford University

Stanford, CA 94305

USA

Matthew Stoffregen

Department of Mathematics

University of California, Los Angeles

Los Angeles, CA 90095

USA

**Abstract:**

The Stern diatomic sequence is closely linked to continued fractions
via the Gauss map on the unit interval, which in turn can be understood
via systematic subdivisions of the unit interval. Higher-dimensional
analogues of continued fractions, called multidimensional continued
fractions, can be produced through various subdivisions of a triangle.
We define triangle partition-Stern sequences (TRIP-Stern sequences for
short) from certain triangle divisions developed earlier by the
authors. These sequences are higher-dimensional generalizations of the
Stern diatomic sequence. We then prove several combinatorial results
about TRIP-Stern sequences, many of which give rise to well-known
sequences. We finish by generalizing TRIP-Stern sequences and
presenting analogous results for these generalizations.

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(Concerned with sequences
A000045
A000930
A000931
A002487
A006131
A007689
A061646
A080040
A155020
A200752
A215404
A271485
A271486
A271487
A271488
A271489
A278612
A278613
A278614
A278615 and
A278616.)

Received June 29 2016; revised versions received August 12 2016; December 20 2016.
Published in *Journal of Integer Sequences*, December 26 2016.

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