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[ Old bulletin had the rule transcribed wrong! ]

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TSPatternEvolve2[init_,t_]:=With[{ru=Dispatch[{{0,_,s___}{s,1},{1,_,s___}{s,1,1,0}}]},NestList[Replace[#,ru]&,init,t]]
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ListStepPlot[Length/@TSPatternEvolve2[{1,0},100]]
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ListStepPlot[Differences[Length/@TSPatternEvolve2[{1,0},100]]]
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-1.0
-0.5
0.5
1.0
Note: it cannot eliminate 1s.....
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ListStepPlot[MapIndexed[#1-(Sqrt[2]-1)First[#2]&,Length/@TSPatternEvolve2[{1,0},100]]]
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ListStepPlot[MapIndexed[#1-(Sqrt[2]-1)First[#2]&,Length/@TSPatternEvolve2[{1,0},1000]]]
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-5
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In[]:=
ListStepPlot[MapIndexed[#1-(Sqrt[2]-1)First[#2]&,Length/@TSPatternEvolve2[{1,0},10000]]]
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