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<a[i, 0] a[i, t]>

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Sign/@ResourceFunction["BlockCellularAutomaton"][{{2,2}{1,1},{1,1}{2,2},{1,2}{1,2},{2,1}{2,1},{2,0}{0,2},{1,0}{1,0},{0,2}{2,0},{0,1}{0,1},{0,0}{0,0}},CenterArray[Table[2,20],60],100]

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auto[history_,t0_,dt_]:=Mean[history[[t0]]*history[[t0+dt]]]

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With[{h=Sign/@ResourceFunction["BlockCellularAutomaton"][{{2,2}{1,1},{1,1}{2,2},{1,2}{1,2},{2,1}{2,1},{2,0}{0,2},{1,0}{1,0},{0,2}{2,0},{0,1}{0,1},{0,0}{0,0}},CenterArray[Table[2,20],60],200]},Table[auto[h,t0,dt],{t0,1,100},{dt,1,20}]]

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ListLinePlot[%]

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With[{h=Sign/@ResourceFunction["BlockCellularAutomaton"][{{2,2}{1,1},{1,1}{2,2},{1,2}{1,2},{2,1}{2,1},{2,0}{0,2},{1,0}{1,0},{0,2}{2,0},{0,1}{0,1},{0,0}{0,0}},CenterArray[Table[2,20],60],200]},Table[auto[h,t0,dt],{t0,1,100},{dt,1,20}]]

ac[t0,dt]

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ListLinePlot[Mean[With[{h=Sign/@ResourceFunction["BlockCellularAutomaton"][{{2,2}{1,1},{1,1}{2,2},{1,2}{1,2},{2,1}{2,1},{2,0}{0,2},{1,0}{1,0},{0,2}{2,0},{0,1}{0,1},{0,0}{0,0}},CenterArray[Table[2,20],60],200]},Table[auto[h,t0,dt],{t0,100,170},{dt,1,20}]]]]

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ListLinePlot[Mean[With[{h=Sign/@ResourceFunction["BlockCellularAutomaton"][{{2,2}{1,1},{1,1}{2,2},{1,2}{1,2},{2,1}{2,1},{2,0}{0,2},{1,0}{1,0},{0,2}{2,0},{0,1}{0,1},{0,0}{0,0}},CenterArray[Table[2,20],200],1000]},Table[auto[h,t0,dt],{t0,500,900},{dt,1,50}]]]]

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ListLinePlot[Mean[With[{h=Sign/@ResourceFunction["BlockCellularAutomaton"][{{2,2}{1,1},{1,1}{2,2},{1,2}{1,2},{2,1}{2,1},{2,0}{0,2},{1,0}{1,0},{0,2}{2,0},{0,1}{0,1},{0,0}{0,0}},CenterArray[Table[2,100],500],5000]},Table[auto[h,t0,dt],{t0,500,1800},{dt,1,100}]]]]

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ListLinePlot[Mean[With[{h=Sign/@ResourceFunction["BlockCellularAutomaton"][{{2,2}{1,1},{1,1}{2,2},{1,2}{1,2},{2,1}{2,1},{2,0}{0,2},{1,0}{1,0},{0,2}{2,0},{0,1}{0,1},{0,0}{0,0}},CenterArray[Table[2,100],500],5000]},Table[auto[h,t0,dt],{t0,500,1800},{dt,1,200}]]]]

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ListLinePlot[Mean[With[{h=CellularAutomaton[30,CenterArray[Table[1,100],500],5000]},Table[auto[h,t0,dt],{t0,500,1800},{dt,1,200}]]]]

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