Common Values of Padovan and Perrin Sequences

Abstract

The integer sequence defined by $$P_{n+3}=P_{n+1}+P_{n}$$ P n + 3 = P n + 1 + P n with initial conditions $$P_{0}=P_{1}=P_{2}=1$$ P 0 = P 1 = P 2 = 1 is known as the Padovan sequence $$(P_{n})_{n\in \mathbb {Z}}$$ ( P n ) n ∈ Z . The Perrin sequence $$(R_{m})_{m\in \mathbb {Z}}$$ ( R m ) m ∈ Z satisfies the same recurrence equation as the Padovan sequence but with starting values $$R_{0}=3$$ R 0 = 3 , $$R_{1}=0$$ R 1 = 0 , and $$R_{2}=2$$ R 2 = 2 . In this note, we solve the Diophantine equation $$P_{n}=\pm R_{m}$$ P n = ± R m with $$(n,m)\in \mathbb {Z}^{2}$$ ( n , m ) ∈ Z 2 .