In this notebook, we attempt to unify works related to sequences generated by linear recurrence relations. To this end, we introduce the concepts of normal, derivative, and primitive sequences and study their properties and interconnections. The work also introduces a new class of matrices associated with the partitions of a natural number into natural summands, and with their help, the concept of sequence rank is proposed. This approach allowed us to classify sequences, establish some of their new properties, and construct conjugate normed matrix spaces.
Notes from this notebook will be useful both for enthusiasts and professional mathematicians.