Matrices formula for Padovan and Perrin sequences

Abstract

The Padovan and Perrin numbers have the matrix formula, Qn ⎡ ⎣ 0 3 0 0 1 2 ⎤ ⎦ = ⎡ ⎣ 0 1 0 0 0 1 1 1 0 ⎤ ⎦ n ⎡ ⎣ 0 3 0 0 1 2 ⎤ ⎦ . The matrix product is a 3× 2 matrix that when raised to the nth power give a matrix product whose entries are Padovan and Perrin numbers. For which we established by mathematical induction such that, Q ⎡ ⎣ 0 3 0 0 1 2 ⎤ ⎦ = ⎡ ⎣ 0 1 0 0 0 1 1 1 0 ⎤ ⎦ n ⎡ ⎣ 0 3 0 0 1 2 ⎤ ⎦ = ⎡ ⎣ Pn Pn Pn+1 Pn+1 Pn+2 Pn+2 ⎤ ⎦ , where Pn and Pn are the Padovan and Perrin sequences, respectively. Mathematics Subject Classification: 11B39, 11B99