On Diophantine equations involving intersection of Thabit and Williams numbers base $b$ and some ternary recurrent sequences

Let $\mathcal{P}_{n}$ be the $n$-th Padovan number, $E_{n}$ be the $n$-th Perrin number, and $N_{n}$ be the $n$-th Narayana's cows number. Let $b$ be a positive integer such that $b\geq 2$. In this paper, we study the Diophantine equations $$\mathcal{P}_{n}=(b\pm 1)\cdot b^{l}\pm 1,\qquad E_{n}=(b\pm 1)\cdot b^{l}\pm 1,\qquad \mbox{and} \qquad N_{n}=(b\pm 1)\cdot b^{l}\pm 1,$$ in non-negative integers $n$, $b$, and positive integer $l$. As a result, we determine the Padovan, Perrin and Narayana's cows numbers that are Thabit and Williams numbers base $b$. Moreover, we determine all solutions of the above equations within the range $2\leq b\leq 10$.