Good evening,
I need to determine an elegant (as possible) method of computing an $n\times n$ matrix below (the matrix shown is just a $16\times16$ example). I've been toying with `For` loops, `Table`s, `Band`s, `DiagonalMatrix`, etc. but due to my lack of *Mathematica* knowledge, I'm at a complete loss as to which method would be most computationally efficient.
Any help would be greatly appreciated! Thanks in advance.

\begin{array}{cccccccccccccccc}
 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
 6 & 0 & 12 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
 0 & 16 & 0 & 16 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
 10 & 0 & 20 & 0 & 20 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
 0 & 24 & 0 & 24 & 0 & 24 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
 14 & 0 & 28 & 0 & 28 & 0 & 28 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
 0 & 32 & 0 & 32 & 0 & 32 & 0 & 32 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
 18 & 0 & 36 & 0 & 36 & 0 & 36 & 0 & 36 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
 0 & 40 & 0 & 40 & 0 & 40 & 0 & 40 & 0 & 40 & 0 & 0 & 0 & 0 & 0 & 0 \\
 22 & 0 & 44 & 0 & 44 & 0 & 44 & 0 & 44 & 0 & 44 & 0 & 0 & 0 & 0 & 0 \\
 0 & 48 & 0 & 48 & 0 & 48 & 0 & 48 & 0 & 48 & 0 & 48 & 0 & 0 & 0 & 0 \\
 26 & 0 & 52 & 0 & 52 & 0 & 52 & 0 & 52 & 0 & 52 & 0 & 52 & 0 & 0 & 0 \\
 0 & 56 & 0 & 56 & 0 & 56 & 0 & 56 & 0 & 56 & 0 & 56 & 0 & 56 & 0 & 0 \\
 30 & 0 & 60 & 0 & 60 & 0 & 60 & 0 & 60 & 0 & 60 & 0 & 60 & 0 & 60 & 0 \\
\end{array}