Elementary proofs of various facts about 3-cores

Using elementary means, we derive an explicit formula for a₃(n), the number of 3-core partitions of n, in terms of the prime factorization of 3n+1. Based on this result, we are able to prove several infinite families of arithmetic results involving a₃(n), one of which specializes to the recent result of Baruah and Berndt which states that, for all n ≥ 0, a ₃(4n+1)=a₃(n).