We shall define a square lamina to be a square outline with a square "hole" so that the shape possesses vertical and horizontal symmetry.

Given eight tiles it is possible to form a lamina in only one way: $3\times 3$ square with a $1\times 1$ hole in the middle. However, using thirty-two tiles it is possible to form two distinct laminae.

If t represents the number of tiles used, we shall say that $t=8$ is type $L(1)$ and $t=32$ is type $L(2)$.

Let $N(n)$ be the number of $t\le 1000000$ such that $t$ is type $L(n)$; for example, $N(15)=832$.

What is $\sum _{n=1}^{10}N(n)$?