Linearly recurrent sequences and $S$-adic sequences (Natural extension of arithmetic algorithms and S-adic system)

"Natural extension of arithmetic algorithms and S-adic system". July 20～24, 2015. edited by Shigeki Akiyama. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.The main part of this article is a survey of a complete characterization of linearly recurrent sequences in terms of S-adic sequences which is due to F. Durand (2000, 2003). However, we also discuss almost minimal sequences in connections with S-adic sequences