In this note, we provide three new, very short proofs of two interesting congruences for Merca's partition function a (n), which enumerates integer partitions where the odd parts have multiplicity at most 2. These modulo 2 congruences were first shown elementarily by Sellers. We then frame a (n) into the much broader context of eta-quotients, and suggest how to comprehensively describe its parity behavior. In particular, extensive computations suggest that a (n) is odd precisely 25\% of the time.
Fabrizio Zanello (Thu,) studied this question.