Exactly right. The reason it's a sum of squares is because of the number of positions for the smaller squares along each side. The 7x7 square has two positions, so there are four positions in total. The 6x6 has three positions on each side, so there are nine positions in total.
By using this reasoning, one can easily see that there are
8^3+7^3+6^3+5^3+4^3+3^3+2^3+1^3 cubes in an 8x8x8 cube.
And so on, for higher dimensional objects.
Over to you, HIMS. Just don't ask what the MobileRead secret is!