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

### Problem 174

Published on Saturday, 22nd December 2007, 01:00 pm; Solved by 4021; Difficulty rating: 40%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: 3x3 square with a 1x1 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 ∑ N(`n`) for 1 ≤ `n` ≤ 10?