Representation of $\frac{1}{2}(F_n-1)(F_{n+1}-1)$ and $\frac{1}{2}(F_n-1)(F_{n+2}-1)$

Abstract

Let $a, b\in \mathbb{N}$ be relatively prime. We consider $(a-1)(b-1)/2$, which arises in the study of the $pq$-th cyclotomic polynomial, where $p,q$ are distinct primes. We prove two possible representations of $(a-1)(b-1)/2$ as nonnegative, integral linear combinations of $a$ and $b$. Surprisingly, for each pair $(a,b)$, only one of the two representations exists and the representation is also unique. We then investigate the representations of $(F_n-1)(F_{n+1}-1)/2$ and $(F_n-1)(F_{n+2}-1)/2$, where $F_i$ is the $i^{th}$ Fibonacci number, and observe several nice patterns.