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

Let Pₙ be the n-th Padovan number, Eₙ be the n-th Perrin number and Nₙ be the n-th Narayana's cows number. Let b be a positive integer such that b ≥ 2. In this paper, we study the Diophantine equations \[ Pₙ = (b ± 1)· bˡ ± 1, \] \[ Eₙ = (b ± 1)· bˡ ± 1, \] and \[ Nₙ = (b ± 1)· bˡ ± 1, \] 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 ≤ b ≤ 10.