I won't bet my hand for this but seems to be ok:

    ClearAll[return];
    return[x : {0 ..., 1}, list_] := {#, list[[;; #]]} &[Length@x];
    return[x_, y_] := {Length@y, y};
    
    sqPeriod[list_] := return[FindLinearRecurrence[list], list]



    sqPeriod /@ {
       {19, 6, 19, 6, 19, 6, 19, 6, 19, 6, 19, 6},
       {73, 7, 4, 73, 7, 4, 73, 7, 4, 73, 7, 4, 73, 7},
       {73, 7, 4, 7, 2, 6, 7, 2, 7, 73, 9, 17, 7, 7}
       } // Column
![enter image description here][1]


  [1]: https://i.sstatic.net/zxu9x.png