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MATHEMATICA
| (Mma program from Jacob A. Siehler) a[n_, i_, j_ ] := Table[If[Total[Abs[{i, j} - {r, s}]] <= 1, 1, 0], {r, n}, {s, n}] //Flatten
a[n_, k_ ] := a[n, Quotient[k + n - 1, n], Mod[k, n, 1]]
m[n_ ] := a[n, # ] & /@ Range[n^2]
ker[n_ ] := NullSpace[m[n], Modulus -> 2]
b[n_ ] := Table[1, {n^2}]
sol[n_ ] := LinearSolve[m[n], b[n], Modulus -> 2];
allSolutions[n_ ] := Module[{s, k},
s = sol[n];
k = ker[n];
Mod[(s + # ) & /@ (Total[(#*k)] & /@ Tuples[{0, 1}, Length[k]]), 2]
] //Sort
MatrixRotate[m_ ] := Transpose[Reverse[m]]
MatrixRotate[m_, n_ ] := Nest[MatrixRotate, m, Mod[n, 4]]
DihedralOrbit[m_ ] := Union@Join[
MatrixRotate[m, # ] & /@ Range[0, 3],
MatrixRotate[Reverse[m], # ] & /@ Range[0, 3]
]
essentialSolutions[n_ ] := Module[{as},
as = Partition[ #, n] & /@ allSolutions[n];
Union[as, SameTest -> (MemberQ[DihedralOrbit[ #1], #2] &)]
]
Length[essentialSolutions[ # ]] & /@ Range[16]
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