Magic Constant of Magic Square/Sequence

Sequence of Magic Sums of Magic Squares
The sequence of the magic sums of magic squares of order $n$ begins:
 * $1, (5,) \, 15, 34, 65, 111, 175, 260, 369, 505, \ldots$

However, note that while $5 = \dfrac {2 \left({2^2 + 1}\right)} 2$, a magic square of order $2$ does not actually exist.