On the number of partitions into odd parts or congruent to $\pm 2 \pmod{10}$

Let $R_2(n)$ denote the number of partitions of $n$ into parts that are odd or congruent to $\pm 2 \pmod{10}$. In 2007, Andrews considered partitions with some negative parts and provided a second combinatorial interpretation for $R_2(n)$. In this paper, we give a collection of linear recurrence relations for the partition function $R_2(n)$. As a corollary, we obtain a simple criterion for deciding whether $R_2(n)$ is odd or even. Some identities involving overpartitions and partitions into distinct parts are derived in this context