Summary: 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.

hosoya triangle, generalized Fibonacci number, convolution, non-decreasing Dyck path, Fibonacci binary word