Rule 188

Author

Eric W. Weisstein
August 31, 2016

This notebook downloaded from http://mathworld.wolfram.com/notebooks/CellularAutomata/Rule188.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Rule188.html.

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

Initialization

rule=188;

Binary

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

Plot

V5

ElementaryCADiagram[rule]

V6

ElementaryCADiagram[rule]

V11

RulePlot[CellularAutomaton[188]]

RulePlot[CellularAutomaton[188],{{1},0},15,Mesh->True,PlotLegends->"Icon"]

Variants

ElementaryCAVariants[rule]

Triangle of Values

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

StringJoin@@@Map[ToString,t,{2}]

Flatten[t]

Represented in Decimal

ElementaryCATriangle[rule,15]

FromDigits[#,2]&/@%

FindSequenceFunction[%,n]//FullSimplify

s=(1/120) (-80+5 (((-2))^n)+24 (((-1))^n)+(3+6 I) (((-2 I))^n)+(3-6 I) (((2 I))^n)+105 (2^n));

FullSimplify[Refine[s,Mod[n,2]==0],n∈Integers&&Mod[n,2]==0]

FullSimplify[Refine[s,Mod[n,2]==1],n∈Integers&&Mod[n,2]==1]

Closed Form

ElementaryCADiagram[rule]

CoefficientList[Series[-((1+3x+4(x^2)+12(x^3)+8(x^4)-8(x^5))/(-1+16(x^4)+(x^2)-16(x^6))),{x,0,30}],x]

FromDigits[#,2]&/@ElementaryCATriangle[rule,30]

<<DiscreteMath`RSolve`

Factor[-((1+3x+4(x^2)+12(x^3)+8(x^4)-8(x^5))/(-1+16(x^4)+(x^2)-16(x^6)))]//TraditionalForm//TeXForm

a=FullSimplify[SeriesTerm[-((1+3x+4(x^2)+12(x^3)+8(x^4)-8(x^5))/(-1+16(x^4)+(x^2)-16(x^6))),{x,0,n}],n∈Integers]

Table[(1/60) (-40-5 (((-2))^n)-12 (((-1))^n)+(6-3 I) (((-2 I))^n)+(6+3 I) (((2 I))^n)+105 (2^n)),{n,0,20}]

FullSimplify[a,Mod[n,2]==0&&n∈Integers]

FullSimplify[a,Mod[n,2]==1&&n∈Integers]

Table[Switch[n,
_?OddQ,(1/30) (-14+3 I (((2 I))^n)+55 (2^n)),
_?EvenQ,(1/15) (-13+3 (((2 I))^n)+25 (2^n))
],{n,0,20}]

Generating Function

LookupFormat[FromDigits[#,2]&/@ElementaryCATriangle[rule,12]]