Rule 50

Author

Eric W. Weisstein
August 31, 2016

This notebook downloaded from http://mathworld.wolfram.com/notebooks/CellularAutomata/Rule50.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Rule50.html.

©2016 Wolfram Research, Inc. except for portions noted otherwise

Initialization

rule=50;

Binary

StringJoin@@(ToString/@IntegerDigits[rule,2,8])

Plot

V5

ElementaryCADiagram[rule]

V6

ElementaryCADiagram[rule]

V11

RulePlot[CellularAutomaton[50]]

RulePlot[CellularAutomaton[50],{{1},0},15,Mesh->True,PlotLegends->"Icon"]

Variants

rules=FromDigits[#,2]&/@Cases[IntegerDigits[Range[0,255],2,8],{_,_,1,1,_,0,1,0}]

Transpose[ElementaryCAVariants/@rules]

Union[Flatten[Rest[%]]]

Show[GraphicsArray[List/@(Graphics[{ElementaryCARules[#,{3,2.2},1.3],Text["rule "<>ToString[#],{0,1},
TextStyle->{FontFamily->"Times",FontSlant->"Italic"}]},
AspectRatio->Automatic,PlotRange->All]&/@rules)]]

Evolution

ArrayPlot[ElementaryCellularAutomaton[rule,250],Frame->False,PixelConstrained->True,ImageSize->1000]

Triangle of Values

ColumnForm[t=ElementaryCATriangle[rule,5],Center]

Flatten[t]

Represented in Decimal

FromDigits[#,2]&/@With[{n=10},MapIndexed[Take[#1,n+#2[[1]]{-1,1}+{2,0}]&,ElementaryCellularAutomaton[250,n]]]

FindSequenceFunction[MapIndexed[{#2[[1]]-1,#1}&,%],n]

IntegerDigits[#,2]&/@%

FullSimplify[Sum[(2^(2k)),{k,0,n}],n∈Integers]

Table[%,{n,0,10}]

Generating Function

gf=Factor[-(1/(-1+5x-4x^2))]

CoefficientList[Series[gf,{x,0,10}],x]

SeriesTerm[gf,{x,0,n}]