ELEMENTARY PROOFS OF TWO CONGRUENCES FOR PARTITIONS WITH ODD PARTS REPEATED AT MOST TWICE

In a recent article on overpartitions, Merca considered the auxiliary function a(n) which counts the number of partitions of n where odd parts are repeated at most twice (and there are no restrictions on the even parts). In the course of his work, Merca proved the following: For all n ≥ 0, a(4n + 2) ≡ 0 (mod 2), and a(4n + 3) ≡ 0 (mod 2). Merca then indicates that a classical proof of these congruences would be very interesting. The goal of this short note is to fulfill Merca's request by providing two truly elementary (classical) proofs of these congruences.