Diophantine equation with balancing-like sequences associated with the Shorey–Tijdeman-type problem

Abstract

Let $\{x_{n}\}_{n \geq 0}$ be the balancing-like sequence defined by $$x_{n+1} = A x_{n} - x_{n-1}$$ with initial terms $x_0 = 0, x_1 = 1$ for $A \geq 3$. In this study, we demonstrate how to find all the solutions of the Diophantine equation,

$$\sum_{i=1}^{3} C_{i}x_{n_{i}} = \sum_{i=4}^{6} C_{i}x_{n_{i}}$$

in fixed integers $A \geq 3$, $n_1 > n_2 > n_3\geq 0, n_4 >n_5 > n_6 \geq 0,$ and $C_{1}x_{n_{1}} \neq C_{4} x_{n_4}$, where $C_{i}; 1 \leq i \leq 6$ are given integers satisfying $C_{1} C_{2} C_{3} \neq 0$.