Parity of sums of partition numbers and squares in arithmetic progressions

In this article, we explore the parity of sums of partition numbers at certain places in arithmetic progressions. In particular, we investigate pairs (a,b)∈N2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(a,b)\in \mathbb {N}^2$$\end{document} for which [inline-graphic not available: see fulltext] if and only if bn+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$bn+1$$\end{document} is a square.