Lights Out

The following GAP code was used to produces these tables, changing the appriopriate arguments.

createE := function(c)
  local M, row, r, i;
  M := [];
  for r in [1..c] do
    row := [];
    for i in [1..c] do
      if (r-1 <= i) and (i <= r+1) then
        row[i] := 1;
      else
        row[i] := 0;
      fi;
    od;
    M[r] := row;
  od;
  return M;
end;

createW := function(c, F)
  local Ec;
  Ec := createE(c);
  return F * BlockMatrix([
    [1, 1, -1*Ec],
    [1, 2, -IdentityMat(c)],
    [2, 1, IdentityMat(c)]
  ], 2, 2);
end;

dimensions := function(c, F)
  local w, dimensions, dimension, n;
  w := createW(c, F);
  dimensions := [0];
  n := 1;
  repeat
    dimension := Size(NullspaceMat((w^n){[1..c]}{[1..c]}));
    Add(dimensions, dimension);
    n := n + 1;
  until dimension = c;
  return dimensions;
end;

F := Z(2);

for column in [1..20] do
  dims := dimensions(column, F);
  Display(dims);
od;
            

Z/2Z

[ 0, 0, 1 ]
[ 0, 1, 0, 2 ]
[ 0, 0, 2, 0, 0, 3 ]
[ 0, 0, 0, 0, 4 ]
[ 0, 1, 1, 3, 0, 2, 0, 4, 1, 1, 0, 4, 0, 1, 1, 4, 0, 2, 0, 3, 1, 1, 0, 5 ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 6 ]
[ 0, 0, 2, 0, 0, 4, 0, 0, 2, 0, 0, 7 ]
[ 0, 1, 0, 2, 0, 1, 6, 2, 0, 1, 0, 2, 0, 7, 0, 2, 0, 1, 0, 2, 6, 1, 0, 2, 0, 1, 0, 8 ]
[ 0, 0, 1, 0, 4, 1, 0, 0, 1, 8, 0, 1, 0, 0, 5, 0, 0, 1, 0, 8, 1, 0, 0, 1, 4, 0, 1, 0, 0, 9 ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10 ]
[ 0, 1, 2, 3, 0, 4, 0, 7, 2, 1, 0, 6, 0, 1, 2, 8, 0, 4, 0, 3, 2, 1, 0, 10, 0, 1, 2, 3, 0, 4, 0, 8, 2, 1, 0, 6, 0, 1, 2, 7, 0, 4, 0, 3, 2, 1, 0, 11 ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12 ]
[ 0, 0, 1, 0, 0, 1, 0, 0, 7, 0, 0, 1, 0, 0, 1, 0, 0, 13 ]
[ 0, 1, 0, 2, 4, 1, 0, 2, 0, 5, 0, 2, 0, 1, 4, 2, 8, 1, 0, 6, 0, 1, 0, 2, 4, 1, 0, 2, 0, 5, 0, 2, 0, 9, 4, 2, 0, 1, 0, 6, 0, 1, 0, 2, 4, 1, 0, 2, 0, 5, 8, 2, 0, 1, 4, 2, 0, 1, 0, 6, 0, 1, 0, 2, 4, 1, 0, 10, 0, 5, 0, 2, 0, 1, 4, 2, 0, 1, 0, 6, 0, 1, 0, 2, 12, 1, 0, 2, 0, 5, 0, 2, 0, 1, 4, 2, 0, 1, 0, 6, 0, 9, 0, 2, 4, 1, 0, 2, 0, 5, 0, 2, 0, 1, 4, 2, 0, 1, 8, 6, 0, 1, 0, 2, 4, 1, 0, 2, 0, 5, 0, 2, 0, 1, 4, 10, 0, 1, 0, 6, 0, 1, 0, 2, 4, 1, 0, 2, 0, 5, 0, 2, 8, 1, 4, 2, 0, 1, 0, 6, 0, 1, 0, 2, 4, 1, 0, 2, 0, 13, 0, 2, 0, 1, 4, 2, 0, 1, 0, 6, 0, 1, 0, 2, 4, 1, 8, 2, 0, 5, 0, 2, 0, 1, 4, 2, 0, 1, 0, 6, 0, 1, 0, 10, 4, 1, 0, 2, 0, 5, 0, 2, 0, 1, 4, 2, 0, 1, 0, 6, 8, 1, 0, 2, 4, 1, 0, 2, 0, 5, 0, 2, 0, 1, 4, 2, 0, 9, 0, 6, 0, 1, 0, 2, 4, 1, 0, 2, 0, 5, 0, 2, 0, 1, 12, 2, 0, 1, 0, 6, 0, 1, 0, 2, 4, 1, 0, 2, 0, 5, 0, 10, 0, 1, 4, 2, 0, 1, 0, 6, 0, 1, 0, 2, 4, 1, 0, 2, 8, 5, 0, 2, 0, 1, 4, 2, 0, 1, 0, 6, 0, 1, 0, 2, 4, 9, 0, 2, 0, 5, 0, 2, 0, 1, 4, 2, 0, 1, 0, 6, 0, 1, 8, 2, 4, 1, 0, 2, 0, 5, 0, 2, 0, 1, 4, 2, 0, 1, 0, 14 ]
[ 0, 0, 2, 0, 0, 4, 0, 0, 2, 0, 0, 8, 0, 0, 2, 0, 0, 4, 0, 0, 2, 0, 0, 15 ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16 ]
[ 0, 1, 1, 3, 0, 2, 6, 4, 1, 1, 0, 4, 0, 13, 1, 4, 0, 2, 0, 3, 7, 1, 0, 5, 0, 1, 1, 15, 0, 2, 0, 4, 1, 1, 6, 4, 0, 1, 1, 4, 0, 14, 0, 3, 1, 1, 0, 5, 6, 1, 1, 3, 0, 2, 0, 16, 1, 1, 0, 4, 0, 1, 7, 4, 0, 2, 0, 3, 1, 13, 0, 5, 0, 1, 1, 3, 6, 2, 0, 4, 1, 1, 0, 16, 0, 1, 1, 4, 0, 2, 6, 3, 1, 1, 0, 5, 0, 13, 1, 3, 0, 2, 0, 4, 7, 1, 0, 4, 0, 1, 1, 16, 0, 2, 0, 3, 1, 1, 6, 5, 0, 1, 1, 3, 0, 14, 0, 4, 1, 1, 0, 4, 6, 1, 1, 4, 0, 2, 0, 15, 1, 1, 0, 5, 0, 1, 7, 3, 0, 2, 0, 4, 1, 13, 0, 4, 0, 1, 1, 4, 6, 2, 0, 3, 1, 1, 0, 17 ]
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