Rule 250

Author

Eric W. Weisstein
August 31, 2016

This notebook downloaded from http://mathworld.wolfram.com/notebooks/CellularAutomata/Rule250.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Rule250.html.

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

Initialization

rule=250;

Binary

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

Plot

V5

ElementaryCADiagram[rule]

V6

ElementaryCADiagram[rule]

V11

RulePlot[CellularAutomaton[250]]

RulePlot[CellularAutomaton[250],{{1},0},15,Mesh->True,PlotLegends->"Icon"]

Same

rules=Select[Range[0,255],IntegerDigits[#,2,8][[{3,4,6,7,8}]]=={1,1,0,1,0}&]

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]

Additive Animation

<<GIF`

Export[gifs<>"rule"<>ToString[rule]<>".gif",movie=Show/@With[{g=32},
Table[
RasterGraphics[
CellularAutomaton[rule,{
ReplacePart[
Table[0,{2g+1}],1,{{g+1},{s+g+1}}
],0},g,{All,{0,2g+1}}
],
Mesh->False
],
{s,-g,g}]
]]//Timing

Show[GraphicsArray[Partition[Take[movie,9],3]]]

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[%,n]

IntegerDigits[#,2]&/@%

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

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