MacMahon's count of 3x3 magic squares with line sum a multiple of 3

MacMahon's 1915 count of $3 \times 3$ magic squares whose line sum is a multiple of $3$.

Write $M_{3}(t)$ for the number of $3 \times 3$ arrays of nonnegative integers whose three rows, three columns and two main diagonals all sum to $t$ (entries need not be distinct). Then $M_{3}(t)$ vanishes unless $3 \mid t$, and for $t = 3e$,

which is the integral form of $\tfrac{2}{9}t^{2} + \tfrac{2}{3}t + 1$.

The companion statement MagicSquares.magic_count_three_otherwise records the vanishing when $3 \nmid t$; together they give the complete counting function.

Formalization Note magicCount n t counts arrays with entries in Fin (t+1) satisfying the magic identities after coercion to ℕ. This is lossless because every entry of a nonnegative magic square with line sum $t$ is at most $t$.