Akademik Veri Yönetim Sistemi
Integers, cilt.26, sa.A5, ss.1-19, 2026 (Scopus)
In this paper, we consider an $mr$-th order linear recurring sequence ${s_n}$ over $\mathbb{R}$, which is a generalization of the general conditional sequences defined by Panario et al. in \cite{genconseq}. The sequence ${s_n}$ contains infinitely many sequences that are not included in the conditional sequences, for example, inverses of cyclotomic polynomials, the Kronecker symbol, etc. We calculate the Binet form of the sequence ${s_n}$ by using $r$-decimation sequences, that is, the sequence obtained by taking every $r$-th term of ${s_n}$. The paper also provides the Binet form of the general conditional sequences for given $m$ and $r$ by using the successor operator method. Finally, some open problems and conjectures are given. One of the conjectures presents the relationship among the successor operator, the coefficients of the linear recurrence relations satisfied by the conditional sequences, and the integer partitions.