Why do I need a second | at the end of my custom column description

I just posted recently, it was marked duplicate, and a lot of the errors I made were corrected. Now, an updated question and the context

One issue I just noticed is that the tikzpicture isn't centered in the minipage in the table. How can I resolve this?

Using the below code, I noticed that I need to include a second | at the end of my column definition. If I don't, then I won't have the right column line. Unsure why?

Additionally, I'd like to know if there's a way to set the padding for the tikzpicture's minipage environment and the left/right equation column pair to be the same, so that the table size doesn't have to be manually controlled?

Context: I'm trying to make a two column table with a tikzpicture on the left and an equation on the right, aligned at the equality symbols. Utilizing 1, 2, and 3, I have arrived at the following:

\documentclass{article}
\usepackage{tikz,array,longtable,calc} % Not 100% sure if I actually need calc?
\newcolumntype{M}[1]
{
>{
\begin{minipage}{#1\textwidth-1.5\arrayrulewidth-2\tabcolsep}
\vspace{\tabcolsep}
\centering
}
c
<{
\vspace{\tabcolsep}
\end{minipage}
}
}
\newcolumntype{A}[1]
{%
>{$} r <{$}
@{\extracolsep{0pt}}
>{${}} l <{$}
}%
\begin{document}
\begin{longtable}{| M{0.22} | A ||}
\hline
\begin{tikzpicture}[baseline=(current bounding box.center)]
\draw [gray] (0,0) circle (15pt);
\filldraw [gray] (0,0) circle (1pt);
\draw [gray] (37.5pt,0) circle (15pt);
\filldraw [gray] (37.5pt,0) circle (1pt);
\end{tikzpicture} & \underline{2r_c \sin \left( ^{\pi} \!/ _{n_k} \right)}  &\geq  2r_f + R_1 r_f
\tabularnewline
\hline
\begin{tikzpicture} [baseline=(current bounding box.center)]
\draw [gray] (0,0) circle (15pt);
\filldraw [gray] (0,0) circle (1pt);
\draw [gray] (30pt,0) circle (15pt);
\filldraw [gray] (30pt,0) circle (1pt);
\end{tikzpicture} & 2r_c \sin \left( ^{\pi} \!/ _{n_k} \right)  &\leq   2r_f+\underline{R_1 r_f} \tabularnewline
\hline
\begin{tikzpicture} [baseline=(current bounding box.center)]
\draw [gray] (0,0) circle (15pt);
\filldraw [gray] (0,0) circle (1pt);
\draw [gray] (22.5pt,0) circle (15pt);
\filldraw [gray] (22.5pt,0) circle (1pt);
\end{tikzpicture} & 2r_c \sin \left( ^{\pi} \!/ _{n_k} \right)  &<   \underline{2r_f}+R_1 r_f \tabularnewline
\hline
\begin{tikzpicture} [baseline=(current bounding box.center)]
\draw [gray] (0,0) circle (15pt);
\filldraw [gray] (0,0) circle (1pt);
\draw [gray] (11.25pt,0) circle (15pt);
\filldraw [gray] (11.25pt,0) circle (1pt);
\end{tikzpicture} & 2r_c \sin \left( ^{\pi} \!/ _{n_k} \right)  &<   \underline{2r_f}+R_1 r_f \tabularnewline
\hline
\end{longtable}
\end{document}

• In my continued efforts to properly center the tikz/minipage/longtable entry bonanza, I have arrived at the following and cannot find the options or lengths that affect the top or left margins of the graphics. \documentclass{article} – Ronen Dec 5 '17 at 23:27

\newcolumntype{A}[1]

so your A column takes an argument (the first |) but discards it as there is no #1 in the definition.