A Pair of Diophantine Equations and Fibonacci-Like Sequences

Given two relatively prime numbers $a$ and $b$, it is known that exactly one of the two Diophantine equations has a nonnegative integral solution $(x,y)$: $$ax+by=\frac{(a-1)(b-1)}{2}and1+ax+by=\frac{(a-1)(b-1)}{2}\mathrm{.}$$ Furthermore, the solution is unique. This paper surveys recent results on finding the solution and determining which equation is used when $a$ and $b$ are taken from certain sequences. We contribute to the literature by finding $(x,y)$ when $a$ and $b$ are consecutive terms of sequences having the Fibonacci recurrence and arbitrary initial terms.