On a Pair of Diophantine Equations

For relatively prime natural numbers $a$ and $b$, we study the two equations $ax+by=(a-1)(b-1)\mathrm{/}2$ and $ax+by+1=(a-1)(b-1)\mathrm{/}2$, which arise from the study of cyclotomic polynomials. Previous work showed that exactly one equation has a nonnegative solution, and the solution is unique. Our first result gives criteria to determine which equation is used for a given pair $(a,b)$. We then use the criteria to study the sequence of equations used by the pair $(a_n\mathrm{/}\gcd (a_n,a_{n+1}),a_{n+1}\mathrm{/}\gcd (a_n,a_{n+1}))$ from several special sequences $(a_n{)}_{n\ge 1}$. Finally, fixing $k\in \mathbb{N}$, we investigate the periodicity of the sequence of equations used by the pair $(k\mathrm{/}\gcd (k,n),n\mathrm{/}\gcd (k,n))$ as $n$ increases.