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:

$1, \paren {10,} \, 45, 136, 325, 666, 1225, 2080, 3321, 5050, 7381, 10 \, 440, 14 \, 365, 19 \, 306, 25 \, 425, 32 \, 896, \ldots$

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

This sequence is A037270 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).