Has the number $f(n)$ of $n \times n$, $0{-}1$ matrices whose determinant is $+1$ been enumerated? E.g., for $n{=}2$, there are $f(2)=3$ such matrices: $$ \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \\ \end{array} \right) \;,\; \left( \begin{array}{cc} 1 & 0 \\ 1 & 1 \\ \end{array} \right) \;,\; \left( \begin{array}{cc} 1 & 1 \\ 0 & 1 \\ \end{array} \right) \;. $$ For $n{=}3$, I count $f(3)=84$ such matrices, from $$ \left( \begin{array}{ccc} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\ \end{array} \right) \;, $$ to $$ \left( \begin{array}{ccc} 1 & 1 & 1 \\ 1 & 1 & 0 \\ 0 & 1 & 1 \\ \end{array} \right) \;. $$

These matrices are a subset of $SL(n,\mathbb{R})$.

**Update**. Oh, I see $f(n)$ is OEIS A086264: $$ 1, 3, 84, 10020, 4851360, 9240051240. $$ No substantive information is provided in OEIS besides those six computed values.

**Addendum**. Unrevealing, but just as a curiosity, here is an overlay of the $84$ equal-volume parallelepipeds that result by applying the $n{=}3$ matrices to the $3 \times 2 \times 1$ box with lowerleft corner at the origin:

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