Magic Constant of Magic Square/Sequence

Sequence of Magic Constants of Magic Squares

The sequence of the magic constants of magic squares of order $n$ begins:

$1, (5,) \, 15, 34, 65, 111, 175, 260, 369, 505, 671, 870, 1105, 1379, 1695, 2056, 2465, 2925, 3439, \ldots$

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

This sequence is A006003 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).