A Discrete Convolution on the Generalized Hosoya Triangle
Department of Mathematics
University of South Carolina
Columbia, SC 29208
Department of Mathematics and Computer Science
Charleston, SC 29409
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.
Full version: pdf,
(Concerned with sequences
August 24 2014;
revised versions received September 16 2014; December 6 2014.
Published in Journal of Integer Sequences, January 8 2015.
Journal of Integer Sequences home page