f[i_, j_] := 1 + Mod[i + j, 4];
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[8]] (* 8x8 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
{n, 1, 12}, {i, 1, n}]] (*
A204255 *)
Join[{1}, Table[Permanent[m[n]], {n, 1, 20}]] (*
A204256 *)