Rule 94

Author

Eric W. Weisstein
August 31, 2016

This notebook downloaded from http://mathworld.wolfram.com/notebooks/CellularAutomata/Rule94.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Rule94.html.

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

Initialization

rule=94;

Binary

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

Plot

V5

ElementaryCADiagram[rule]

V6

ElementaryCADiagram[rule]

V11

RulePlot[CellularAutomaton[94]]

RulePlot[CellularAutomaton[94],{{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,20]

FromDigits[#,2]&/@%

FindSequenceFunction[%,n]

FunctionExpand[%]

Table[DifferenceRoot[Function[{\[FormalY],\[FormalN]},{30-16 \[FormalY][\[FormalN]]-16 \[FormalY][1+\[FormalN]]+\[FormalY][2+\[FormalN]]+\[FormalY][3+\[FormalN]]==0,\[FormalY][1]==1,\[FormalY][2]==7,\[FormalY][3]==27,\[FormalY][4]==119,\[FormalY][5]==427}]][n],{n,10}]

Table[(1/60) (60+(((-4))^n)-40 (((-1))^n)-(2^(1+2 n))-(((-1))^n) (2^(1+2 n))+(4^n) (107-4 (((-1))^n))),{n,10}]

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

FullSimplify[3+(2^(1+2 n))+(2^(2+2 n))-(1/3) (2^(2-Mod[n,2])) (-(2^n)+(2^(Mod[n,2]))) ((2^n)+(2^(Mod[n,2])))/.n->n-1,n∈Integers]

FullSimplify[(1/6) (18+(2^(n+2 Floor[((1+n)/2)]))-(2^(3+Mod[1+n,2]))+9 (4^n)),n∈Integers&&n>2]

Table[Floor[(1+n)/2],{n,5}]

Table[(n+Mod[n,2])/2,{n,5}]

FullSimplify[(1/6) (18+(2^(n+2(n+Mod[n,2])/2))-(2^(3+Mod[1+n,2]))+9 (4^n)),n∈Integers&&n>2]

FullSimplify[(1/6) (18+(2^(2 n+Mod[n,2]))-(2^(3+1-Mod[n,2]))+9 (4^n))]

FullSimplify[(1/6) (18+(2^(2 n+Mod[n,2]))-(2^(3+1-Mod[n,2]))+9 (4^n)),Mod[n,2]==0]

FullSimplify[(1/6) (18+(2^(2 n+Mod[n,2]))-(2^(3+1-Mod[n,2]))+9 (4^n)),Mod[n,2]==1]

If[OddQ[n],(1/3) (1+5 (4^n)),(1/6) (10+11 (4^n))]

IntegerDigits[#,2]&/@Table[If[EvenQ[n],(1/3) (1+5 (4^n)),(1/6) (10+11 (4^n))],{n,0,6}]

IntegerDigits[#,2]&/@Table[(1/6) (18+(2^(2 n+Mod[n,2]))-(2^(3+1-Mod[n,2]))+9 (4^n)),{n,0,6}]

FromDigits[#,2]&/@%

Table[Switch[n,
0,1,
1,7,
_?OddQ,(1/6) (10+11 (4^n)),
_?EvenQ,(1/3) (1+5 (4^n))],{n,0,20}]

FromDigits[#,2]&/@With[{n=20},MapIndexed[Take[#1,n+#2[[1]]{-1,1}+{2,0}]&,ElementaryCellularAutomaton[rule,n]]]

Generating Function

gf=Factor[-((-1-7x-10(x^2)+16(x^4)+32(x^5))/(1-17(x^2)+16(x^4)))]

-(((1+2 x) (-1-5 x+16 (x^4)))/((-1+x) (1+x) (-1+4 x) (1+4 x)))//TraditionalForm//TeXForm

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

FullSimplify[SeriesTerm[gf,{x,0,n}],n≥0&&n∈Integers]

FullSimplify[SeriesTerm[gf,{x,0,n}],n≥2&&n∈Integers]

FullSimplify[SeriesTerm[gf,{x,0,n}],n≥2&&Mod[n,2]==0]

FullSimplify[SeriesTerm[gf,{x,0,n}],n≥2&&Mod[n,1]==0]