Matrix Representation of Recursive Sequences of Order 3 and Its Applications

Abstract Here presented is a matrix representation of recursive number sequences of order 3 defined by an = pan−1 + qan−2 + ran−3 with arbitrary initial conditions a0, a1 = 0, and a2 and their special cases of Padovan number sequence and Perrin number sequence with initial conditions a0 = a1 = 0 and a2 = 1 and a0 = 3, a1 = 0, and a2 = 2, respectively. The matrix representation is used to construct many well known and new identities of recursive number sequences as well as Pavodan and Perrin sequences.