Matrix Representation of Recursive Sequences of Order 3 and Its Applications

Here presented is a matrix representation of recursive number sequences of order 3 defined by a n = pa n - 1 + qa n - 2 + ra n - 3 with arbitrary initial conditions a 0 ; a 1 = 0, and a 2 and their special cases of Padovan number sequence and Perrin number sequence with initial conditions a 0 = a 1 = 0 and a 2 = 1 and a 0 = 3, a 1 = 0, and a 2 = 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.