\documentclass[tikz]{standalone}
 
\begin{document}
    \begin{tikzpicture}
        \tikzset{dot/.style={minimum size=5.5pt, draw=white, line width=.5pt, fill=black, circle, inner sep=0pt}}
        \clip (0.5, 0.5) rectangle (13.5, 9.5);
        \draw[gray!90, line width=3pt] (0.5, 0.5) grid (13.5, 9.5);
        \draw[gray!90, step=sqrt(2), line width=3pt, rotate=-45] (-7, 0.5) grid (9.5, 16.5);
 
        \foreach \x in {1,5,...,13} {
            \foreach \y in {1,5,...,9} {
                \node[dot] at (\x,\y) {};
            }
        }
    \end{tikzpicture}
\end{document}