Defining a new bracket style

The MnSymbol package has your desired symbols as \langlebar and \ranglebar. They also work with \left and \right.

\documentclass{article}
\usepackage{MnSymbol}
\begin{document}
\[\mu(\langlebar a \ranglebar_{n}) = \sum_{k = 0}^{p-1} \mu (\langlebar a + kp^n \ranglebar_{n+1})\]
\end{document}

As you can see it also changes other symbols. If you don't want to use the entire MnSymbol package but only these brackets, you have to include the relevant parts in your document.

\documentclass{article}
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}
\DeclareFontShape{OMX}{MnSymbolE}{b}{n}{
    <-6>  MnSymbolE-Bold5
   <6-7>  MnSymbolE-Bold6
   <7-8>  MnSymbolE-Bold7
   <8-9>  MnSymbolE-Bold8
   <9-10> MnSymbolE-Bold9
  <10-12> MnSymbolE-Bold10
  <12->   MnSymbolE-Bold12}{}

\DeclareSymbolFont{MnSymbol}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{MnSymbol}{bold}{OMX}{MnSymbolE}{b}{n}

\DeclareMathDelimiter{\langlebar}{\mathopen}{MnSymbol}{'152}{MnSymbol}{'152}
\DeclareMathDelimiter{\ranglebar}{\mathclose}{MnSymbol}{'157}{MnSymbol}{'157}
\begin{document}
\[\mu(\langlebar a \ranglebar_{n}) = \sum_{k = 0}^{p-1} \mu (\langlebar a + kp^n \ranglebar_{n+1})\]
\end{document}

---

In unicode-math such a symbol is available but not in every font. We have to load it using the range option from, e.g., XITS Math.

\documentclass{article}
\usepackage{unicode-math}
\setmathfont{Latin Modern Math}
\setmathfont[range={\llangle,\rrangle}]{XITS Math}
\begin{document}
\[\mu(\llangle a \rrangle_{n}) = \sum_{k = 0}^{p-1} \mu (\llangle a + kp^n \rrangle_{n+1})\]
\end{document}

While I use scalerel to scale the symbols, I only scale them to the size of [ and ], rather than with arbitrary stretch. Here, I just use \triangleleft and \triangleright as the base shapes.

EDITED to use \mathopen and \mathclose, instead of \mathinner.

\documentclass[11pt]{article}
\usepackage{amsmath,scalerel}
\newcommand\langlebar{\mathopen{\scalerel*[.55ex]{\triangleleft\mkern3mu}{[}}}
\newcommand\ranglebar{\mathclose{\scalerel*[.55ex]{\mkern3mu\triangleright}{]}}}
\begin{document}
\[\mu(\langlebar a\ranglebar_n) = \sum_{k = 0}^{p-1} \mu (
  \langlebar a + kp^n\ranglebar_{n+1})\] 
\end{document}