ClearAll[KolakoskiGen]
KolakoskiGen[start_List, its_Integer] := Module[{c, s, k, cnext, sk},
  s = {};
  k = 1;
  c = start;
  Do[
   cnext = First[c];
   c = RotateLeft[c];
   AppendTo[s, cnext];
   sk = s[[k]];
   If[sk > 1,
    s = Join[s, ConstantArray[cnext, sk - 1]]
    ];
   k += 1;
   ,
   {its}
   ];
  s
  ]

run = Take[KolakoskiGen[{1, 2}, 20], 20]
check = Length /@ Split[%];
check === Take[run, Length[check]]

run = Take[KolakoskiGen[{2, 1}, 20], 20]
check = Length /@ Split[%];
check === Take[run, Length[check]]

run = Take[KolakoskiGen[{1, 3, 1, 2}, 30], 30]
check = Length /@ Split[%];
check === Take[run, Length[check]]