Journal of Integer Sequences, Vol. 20 (2017), Article 17.1.7

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


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.

Full version:  pdf,    dvi,    ps,    latex    

(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.

Return to Journal of Integer Sequences home page