Proof of a conjecture of Ballantine and Merca on truncated sums of 6-regular partitions

In 2012, Andrews and Merca proved a truncated theorem on Euler's pentagonal number theorem. Their work has opened up a new study of truncated series. Recently, Ballantine and Merca posed a conjecture on infinite families of inequalities involving $b_6\left(n\right)$, which counts the number of 6-regular partitions of n. In this paper, we confirm Ballantine and Merca's conjecture on linear equalities of $b_6\left(n\right)$ based on a formula of the number of partitions of n into parts not exceeding 3 due to Cayley.