Bordered magic squares

A bordered magic square is a magic square, which remains magic when its borders are removed. Let us take the depicted magic square of order 8. When we remove its border, we will get a square of order 6.

6449546331053
601516474849205
744224241252158
5133372930283814
632343536312759
826402423433957
5245461817195013
12615611262551
151647484920
442242412521
333729302838
323435363127
264024234339
454618171950

This square isn't normalized anymore, because it doesn't contain the numbers 1, 2, … , n2 anymore. But still each row, each column and both diagonals sum S=195.

A second condition for bordered magic squares demands that the numbers of the border enclose the numbers of the inner square. Satisfiying this condition means that the numbers 1, … ,2(n−1) and n2−2(n−1) + 1 , … , n2 must form the border. All other numbers must be elements of the inner square.

Let's take a bordered magic square of order n=6 as an example. The inner square is of order n=4, and must be formed of 42=16 numbers. On the other side, there are

62 − 42 = 36 − 16 = 20

numbers for the border, which have to enclose the inner elements. So, ten numbers are less and ten numbers are greater than the inner numbers.

lower numbers of the border: 1 … 10
inner numbers: 11 … 26
upper numbers of the border: 27 … 36

One example of such an arrangement is shown in the following magic square of order 6:

362373231
42613122333
91520211828
271916172210
29142524118
635343051