Rule 190

Author

Eric W. Weisstein
August 31, 2016

This notebook downloaded from http://mathworld.wolfram.com/notebooks/CellularAutomata/Rule190.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Rule190.html.

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

Initialization

rule=190;

Binary

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

Plot

V5

ElementaryCADiagram[rule]

V6

ElementaryCADiagram[rule]

V11

RulePlot[CellularAutomaton[190]]

RulePlot[CellularAutomaton[190],{{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

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

FindSequenceFunction[%,n]

IntegerDigits[#,2]&/@%

Closed Form

ElementaryCADiagram[rule]

FromDigits[#,4]&/@Reverse[NestList[Most,Flatten[Table[{1,3},{5}]],9]]

IntegerDigits[#,4]&/@%

Sum[(4^(k+1))+3 (4^k),{k,0,2n,2}]

Table[%,{n,0,6}]

FullSimplify[(7/15) (-1+(4^(1+Floor[n]))) (1+(4^(1+Floor[n])))/.n->n/2,n∈Integers]

FullSimplify[Refine[(7/15) (-1+(16^(1+Floor[(n/2)]))),Mod[n,2]==1]]

Table[%,{n,1,6,2}]

1+Sum[(4^(k+1))+3 (4^k),{k,1,2n,2}]

Table[%,{n,0,6}]

FullSimplify[Refine[1+(28/15) (-1+(4^(1+Floor[(1/2) (-1+2 n)]))) (1+(4^(1+Floor[(1/2) (-1+2 n)])))/.n->n/2,Mod[n,2]==0]]

Table[%,{n,0,6,2}]

Table[Switch[n,
_?OddQ,(7/15) (-1+(4^(1+n))),
_?EvenQ,(1/15) (-13+7 (4^(1+n)))],{n,0,10}]

Generating Function

LookupFormat[FromDigits[#,2]&/@ElementaryCATriangle[rule,10]]

gf=Factor[-((-1-3x)/(-(x^2)+1-4x+4(x^3)))]

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

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

Table[(1/15) (-10-3 (((-1))^n)+7 (4^(1+n))),{n,0,20}]