Matrices & Systems · Superchallenge problem

Define matrices $H_1 = (1)$ and, for each $m$, the $2m\times 2m$ block matrix $H_{2m} = \begin{pmatrix}H_m & H_m\\H_m & -H_m\end{pmatrix}$. So $H_2 = \begin{pmatrix}1 & 1\\1 & -1\end{pmatrix}$, and $H_{16}$ is a $16\times 16$ matrix of $\pm1$s. Find the sum of all $256$ entries of $(H_{16} + 2I)^{-1}$.