Abstract
The notion of broken k-diamond partitions was introduced by Andrews and Paule. Let \(\Delta _k(n)\) denote the number of broken k-diamond partitions of n for a fixed positive integer k. Recently, Paule and Radu conjectured that \(\Delta _3(343n+82)\equiv \Delta _3(343n+278)\equiv \Delta _3(343n+327)\equiv 0\ (\mathrm{mod} \ 7)\). Jameson confirmed this conjecture and proved that \(\Delta _3(343n+229)\equiv 0 \ (\mathrm{mod} \ 7)\) by using the theory of modular forms. In this paper, we prove several infinite families of Ramanujan-type congruences modulo 7 for \(\Delta _3(n)\) by establishing a recurrence relation for a sequence related to \(\Delta _3(7n+5)\). In the process, we also give new proofs of the four congruences due to Paule and Radu, and Jameson.
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