On Diophantine equations involving Thabit and Williams numbers base $b$

Let $b\ge 2$ be a positive integer. Let $r$ and $s$ be two integers with $r\ge 1$, $s\in\{-1, 1\}$ and $\Delta=r^2+4s > 0$, let $\{U_n\}_{n\ge 0}$ be the Lucas sequence given by $U_{n+2}=rU_{n+1}+sU_n$, with $U_0=0$ and $U_1=1$. In this paper, we investigate the solutions of the Diophantine equations $$U_n\pm U_m=(b\pm 1)\cdot b^\ell\pm1,$$ by giving effective bounds for the variables $n$, $m$ and $\ell$ in terms of $b$, $r$ and $s$. Moreover, we solve the above equation in the cases where $2\le b\le 10$, by considering the Fibonacci, Pell and balancing sequences.