Counting the number of "hollow" square laminae that can form one, two, three, ... distinct arrangements

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 $t ≤ 1000000$ such that $t$ is type L($n$); for example, $N(15) = 832$.

What is $\sum N(n)$ for $1 ≤ n ≤ 10$?