Cullen and Woodall numbers in Padovan and Perrin sequences

Let $\{P_n{\}}_{n\ge 0}$ and $\{R_n{\}}_{n\ge 0}$ denote the Padovan and Perrin sequences, both satisfying the recurrence $U_{n+3}=U_{n+1}+U_n$, but with initial values $P_0=P_1=P_2=1$ and $R_0=3$, $R_1=0$, $R_2=2$, respectively. A $\mathit{\text{Cullen}}\mathit{\text{number}}$ is a positive integer of the form $m\cdot 2^m+1$ for some integer $m\ge 1$, while a $\mathit{\text{Woodall}}\mathit{\text{number}}$ is a positive integer of the form $m\cdot 2^m-1$ for some integer $m\ge 1$. In this paper, we determine all Woodall numbers in the Padovan sequence and all Cullen numbers in the Perrin sequence. Specifically, we prove that $1$ and $7$ are the only Woodall numbers in the Padovan sequence, and that $3$ is the only Cullen number in the Perrin sequence.