Infinite families of congruences modulo 7 for broken 3-diamond partitions

The notion of broken k-diamond partitions was introduced by Andrews and Paule. Let Δk(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _k(n)$$\end{document} denote the number of broken k-diamond partitions of n for a fixed positive integer k. Recently, Paule and Radu conjectured that Δ3(343n+82)≡Δ3(343n+278)≡Δ3(343n+327)≡0(mod7)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _3(343n+82)\equiv \Delta _3(343n+278)\equiv \Delta _3(343n+327)\equiv 0\ (\mathrm{mod} \ 7)$$\end{document}. Jameson confirmed this conjecture and proved that Δ3(343n+229)≡0(mod7)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _3(343n+229)\equiv 0 \ (\mathrm{mod} \ 7)$$\end{document} by using the theory of modular forms. In this paper, we prove several infinite families of Ramanujan-type congruences modulo 7 for Δ3(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _3(n)$$\end{document} by establishing a recurrence relation for a sequence related to Δ3(7n+5)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _3(7n+5)$$\end{document}. In the process, we also give new proofs of the four congruences due to Paule and Radu, and Jameson.