On a family of diophantine triples {K,A2K + 2A, (A + 1)2K + 2(A + 1)} with two parameters II

Let A and k be positive integers. In this paper, we study the Diophantine quadruples \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{ k,A^2 k + 2A,(A + 1)^2 k + 2(A + 1)d\} .$$\end{document} If d is a positive integer such that the product of any two distinct elements of the set increased by 1 is a perfect square, then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{gathered} d = (4A^4 + 8A^3 + 4A^2 )k^3 + (16A^3 + 24A^2 + 8A)k^2 + \hfill \\ + (20A^2 + 20A + 4)k + (8A + 4) \hfill \\ \end{gathered} $$\end{document} for A ≥ 52330 and any k. This extends our result obtained in [4].