JOURNAL OF UNIVERSAL MATHEMATICS Vol.7 No.1 pp.1-11 (2024) ISSN-2618-5660 DOI: 10.33773/jum.1372291 A NOTE OF THE COMBINATORIAL INTERPRETATION OF THE PERRIN AND TETRARRIN SEQUENCE RENATA P. M. VIEIRA, ELEN V. P. SPREAFICO, FRANCISCO R. V. ALVES, AND PAULA M. M. C. CATARINO 0000-0002-1966-7097, 0000-0001-6079-2458, 0000-0003-3710-1561, and 0000-0001-6917-5093 Abstract. The present study carries out an investigation around the Per- rin and Tetrarrin numbers, allowing a combinatorial interpretation for these sequences. Furthermore, it is possible to establish a study around the respec- tive polynomial numbers of Perrin and Tetrarrin, using the bracelet method. With this, we have the definition of combinatorial models of these numbers, contributing to the evolution of these sequences with their respective combina- torial approaches. As a conclusion, there is a discussion of theorems referring to the combinatorial models of these sequences, allowing the study of the mathematical advancement of these numbers. 1. Introduction The present word aims to introduce new interpretations for the Perrin sequence, its extension and polynomial forms. In fact, works in the literature containing the existence of recent works are identified, involving new combinatorial approaches of recurrent numerical sequences [1, 2, 3, 4, 8]. With this, it is possible to observe forms of visualization of the terms of these respective studied sequences. Based on this, a combinatorial interpretation is performed for the sequence of Perrin, Tetrarrin and polynomial forms, based on the works of Tedford (2019) [5] and Vieira (2020) [7]. The Perrin sequence is closely related to the Padovan sequence. In a similar way as with the Fibonacci and Lucas sequence. With this, it is worth highlighting the work of Benjamin and Quinn (2003) [1] in which they carried out a study around the combinatorial model of Fibonacci and Lucas, investigating Lucas bracelets. So, the n-bracelet is defined as being a cover of a circular n-board. Lucas sequence has its combinatorial interpretation by means of bracelets, as l n being the number of ways to tile a circular board composed of n cells marked with squares and 1 x 2 1 Date: Received: 2023-10-06; Accepted: 2024-01-29. Key words and phrases. Combinatorics, Perrin sequence, Polynomial sequence.