Linear relations for the number of overpartitions into odd parts

Let p̄ₒ(n) denote the number of overpartitions of n into odd parts. The partition function p̄ₒ(n) has been the subject of many recent studies where many explicit Ramanujan-like congruences were discovered. In this paper, we provide three linear recurrence relation for p̄ₒ(n). Several connections with partitions into parts not congruent to 2 4, overpartitions and partitions into distinct parts are presented in this context.