Padovan squares which are again Padovan

The integer sequence defined by $P_{n+1}=P_{n-1}+P_{n-2}$ with initial values $P_{0}=P_{1}=P_{2}=1$ is known as the Padovan sequence $(P_n)_{n\in \mathbb{Z}}$. In this note, we solve the Diophantine equations $P_{-n}=\pm P_{m}^{2}$, $P_{n}=P_{-m}^{2}$, and $P_{-n}=\pm P_{-m}^{2}$ in positive integers $n,m$.