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A Discrete Convolution on the Generalized Hosoya Triangle
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Éva Czabarka

Department of Mathematics

University of South Carolina

Columbia, SC 29208

USA

Rigoberto Flórez

Department of Mathematics and Computer Science

The Citadel

Charleston, SC 29409

USA

Leandro Junes

Department of Mathematics, Computer Science and Information Systems

California University of Pennsylvania

California, PA 15419

USA

**Abstract:**

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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