Consider an $n\times n$ matrix $M_n$ where the sequence $$\{1,2,3,\dots,n^2\} \mod 4=\{1,2,3,0,1,2,3,\dots\}$$ forms a clock-wise spiral, in that given order. For example, $$M_4=\begin{bmatrix} 1&2&3&0\\ 0&1&2&1\\ 3&0&3&2 \\ 2&1&0&3 \end{bmatrix} \qquad \text{and} \qquad M_5=\begin{bmatrix} 1&2&3&0&1\\ 0&1&2&3&2 \\ 3&0&1&0&3 \\ 2&3&2&1&0 \\ 1&0&3&2&1 \end{bmatrix}.$$

Question. Is it true that $$\det(M_{2n})=3(2n-1)4^{n-1} \qquad \text{and} \qquad \det(M_{2n+1})=-(3n^2-1)4^n\,\,\,?$$

Added clarification. To understand the construction of the above matrices, take a look at the matrices from my other MO question. Then, reduce the entries modulo $4$ and follow through by computing the determinants.