Ducci sequences of triples and continued fractions

Abstract

Using continued fraction expansions, we provide an explicit expression for the orbit of a real vector under the Ducci map. This expression enables us to give an alternative proof of Brockman and Zerr’s characterization theorem on the behaviour of the Ducci sequences on [Fibonacci Quart. 45 (2007), pp. 155–163]. We then consider Ducci matrix sequences, a concept introduced by Hogenson et al. [Linear Algebra Appl. 437 (2012), pp. 285–293], and provide a new proof of their main theorem. Moreover, utilizing the explicit expression, we compute, for two given distinct starting vectors in , the length of a maximal common initial segment of their Ducci matrix sequences. Finally, answering a question by Hogenson et al., we prove that for a given positive irrational number , the Ducci matrix sequence associated with is eventually periodic if and only if is quadratic.