Journal of Integer Sequences, Vol. 18 (2015), Article 15.1.6

A Discrete Convolution on the Generalized Hosoya Triangle

Éva Czabarka
Department of Mathematics
University of South Carolina
Columbia, SC 29208

Rigoberto Flórez
Department of Mathematics and Computer Science
The Citadel
Charleston, SC 29409

Leandro Junes
Department of Mathematics, Computer Science and Information Systems
California University of Pennsylvania
California, PA 15419


The generalized Hosoya triangle is an arrangement of numbers where each entry is a product of two generalized Fibonacci numbers. We define a discrete convolution C based on the entries of the generalized Hosoya triangle. We use C and generating functions to prove that the sum of every k-th entry in the n-th row or diagonal of generalized Hosoya triangle, beginning on the left with the first entry, is a linear combination of rational functions on Fibonacci numbers and Lucas numbers. A simple formula is given for a particular case of this convolution. We also show that C summarizes several sequences in the OEIS. As an application, we use our convolution to enumerate many statistics in combinatorics.

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(Concerned with sequences A001629 A001870 A004798 A004799 A030267 A054444 A056014 A060934 A061171 A094292 A099924 A129720 A129722 A203573 A203574.)

Received August 24 2014; revised versions received September 16 2014; December 6 2014. Published in Journal of Integer Sequences, January 8 2015.

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