Diophantine Equation with Balancing-like Sequences Associated to the Pillai-Tijdeman-type Problem

Let $\{x_n{\}}_{n\ge 0}$ be the balancing-like sequence defined by $x_{n+1} = A x_{n} - x_{n-1}$,for$A>2$,where$x_0 = 0$and$x_1 = 1$. In this paper, we demonstrate how to find all the solutions of the Diophantine equation, $C_{1}x_{n_{1}} + C_{2}x_{n_{2}} + C_{3}x_{n_{3}} = C_{4}x_{n_4} + C_{5}x_{n_5}

• C_{6}x_{n_{6}}$,infixedinteger$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_{1}, C_{2}, C_{3}, C_{4}, C_{5}, C_{6}$aregivenintegerssuchthat$C_{1} C_{2} C_{3} \neq 0$.