Rule 54

Author

Eric W. Weisstein
August 31, 2016

This notebook downloaded from http://mathworld.wolfram.com/notebooks/CellularAutomata/Rule54.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Rule54.html.

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

Initialization

rule=54;

Binary

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

Plot

V5

ElementaryCADiagram[rule]

V6

ElementaryCADiagram[rule]

V11

RulePlot[CellularAutomaton[54]]

RulePlot[CellularAutomaton[54],{{1},0},15,Mesh->True,PlotLegends->"Icon"]

Variants

ElementaryCAVariants[rule]

Evolution

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

Triangle of Values

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

FromDigits/@t

Flatten[t]

Represented in Decimal

ElementaryCATriangle[rule,10]

FromDigits[#,2]&/@%

s=FindSequenceFunction[%,n]//FullSimplify

Refine[s,Mod[n,2]==1]//Simplify

Refine[s,Mod[n,2]==0]//Simplify

Table[(1/30) (-8+3 (((-4))^n)-6 (((-1))^n)+11 (4^n)),{n,10}]

Closed Form

Sum[(2^(4k)),{k,0,n/2}]//Simplify

Table[(1/15) (-1+(2^(2+n))) (1+(2^(2+n))),{n,0,10,2}]

Sum[(2^(4k))+(2^(4k+1))+(2^(4k+2)),{k,0,(n-1)/2}]//Simplify

Table[(7/15) (-1+(2^(1+n))) (1+(2^(1+n))),{n,1,10,2}]

Table[Switch[n,
_?OddQ,(7/15) (-1+(4^(1+n))),
_?EvenQ,(1/15) (-1+(4^(2+n)))],{n,0,10}]

Generating Function

ogf=Factor[-((1+7x)/(-1+17(x^2)-16(x^4)))]

((1+7 x)/((-1+x) (1+x) (-1+4 x) (1+4 x)))//TraditionalForm//TeXForm

CoefficientList[Series[ogf,{x,0,20}],x]

Table[Switch[n,
_?OddQ,(7/15) (-1+(4^(1+n))),
_?EvenQ,(1/15) (-1+(4^(2+n)))],{n,0,20}]