NÚMEROS REPUNIDADES E A REPRESENTAÇÃO MATRICIAL: Números repunits y la representación matricial.

Números repunits y la representación matricial.

Abstract

In this work, we conducted an investigation on the generating matrices of the repunit
sequence. We presented Proposition 3.1, which describes the sequence of R_n in terms of powers of the matrix R. Furthermore, as a result of this investigation, we demonstrated several identities, with a particular emphasis on the Cassini Identity discussed in Corollary 3.5. Meanwhile, Proposition 3.6 exhibits the recursive form for the matrix R, similar to the relationship seen in Equation 1.3. Finally, we provided a proof for Proposition 4.3, establishing an expression for the determinant of a tridiagonal matrix. Consequently, Corollary 4.4 and Proposition 4.6 demonstrate that the recursive form of the repunit sequence arises as a consequence of calculating the determinant of a tridiagonal matrix, as proposed in Proposition 4.3.

Keywords: Recursive Linear Sequences; Repunit Matrix; Repunit Numbers; Tridiagonal Matrices