Diophantine quadruples with values in the Padovan and Perrin sequences

The Padovan (Pn)n≥0 and Perrin (Rn)n≥0 sequences are third-order linear recurrences, both defined by the relation un = un−2+un−3 for n ≥ 3. They differ in their initial conditions resulting in different sequences. The Padovan sequence begins with P0 = P1 = P2 = 1, whereas the Perrin sequence starts with R0 = 3, R1 = 0, and R2 = 2. Motivated by the work of Gómez and Luca [Tribonacci Diophantine quadruples, Glas. Mat., Ser. III, 50(1):17–24, 2015], we investigate whether there exist quadruples of positive integers a1 < a2 < a3 < a4 such that all pairwise products aiaj + 1 (for i ≠ j) belong to the Padovan or Perrin sequence, and we prove that the answer is negative.