Punjab University Journal of Mathematics (ISSN 1016-2526) Vol. 52(11)(2020) pp. 1-9 The (s, t)-Padovan Quaternions Matrix Sequence Renata Passos Machado Vieira 1,* , Francisco Regis Vieira Alves 2 and Paula Maria Machado Cruz Catarino 3 1,2 Department of Mathematics, Federal Institute of Education, Science and Technology of the State of Ceara, Brazil. 3 Department of Mathematics, University of Tras-os-Montes and Alto Douro, Portugal. Corresponding author: Email: * re.passosm@gmail.com Email: 2 fregis@gmx.fr Email: 3 pcatarin@utad.pt Received: 17 April, 2020 / Accepted: 06 October, 2020 / Published online: 12 November, 2020 Abstract.: In this paper, we will introduce the (s, t)-Padovan quaternions matrix sequence. Starting the studies based on the generalization of the Padovan quaternion coefficients in relation to their recurrence, their matrix sequence is then defined. Some mathematical theorems are discussed and the Binet formula and the generating function of this matrix sequence are studied. AMS (MOS) Subject Classification Codes: 11B37; 11B39 Key Words: matrix sequence; Padovan; quaternions. 1. I NTRODUCTION Padovan sequence is a linear and recurring sequence of integers, defined by recurrence: P n = P n-2 + P n-3 ,n  3. With initial values equal to P 0 = P 1 = P 2 =1. Being a third order sequence, it has its characteristic equation given by x 3 − x − 1=0, with three roots, one real and two complex. Its historical process can be found in some arti- cles found in the literature, explaining the construction of this sequence and its relationship exists with the plastic number [15] [10] [16]. In order to study the matrix form, the matrices studied in the works of Sokhuma [12] and Seenukul [11] are taken into account. We also highlight that there are five more Padovan matrices. Thus we have the matrix form of the Padovan sequence given by: 1