Associative magic square

An associative magic square is a magic square for which each pair of numbers symmetrically opposite to the center sum up to the same value.

For an n × n square, filled with the numbers from 1 to n2, this common sum must equal n2 + 1.

[4] The 4 × 4 magic square from Albrecht Dürer's 1514 engraving Melencolia I – also found in a 1765 letter of Benjamin Franklin – is also associative, with each pair of opposite numbers summing to 17.

[5] The numbers of possible associative n × n magic squares for n = 3,4,5,..., counting two squares as the same whenever they differ only by a rotation or reflection, are: The number zero for n = 6 is an example of a more general phenomenon: associative magic squares do not exist for values of n that are singly even (equal to 2 modulo 4).

[3] Every associative magic square of even order forms a singular matrix, but associative magic squares of odd order can be singular or nonsingular.