Eccoci.

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 $\sum\limits_{n = 1}^{10} N(n)$?