Sum of Terms of Magic Square/Sequence

Sequence of Sums of Terms of Magic Squares
The sequence of the sum totals of all the entries in magic squares of order $n$ begins:
 * $1, \paren {10,} \, 45, 136, 325, 666, 1225, \ldots$

However, note that while $10 = \dfrac {2^2 \paren {2^2 + 1} } 2$, a magic square of order $2$ does not actually exist.