Rule 62

Author

Eric W. Weisstein
August 31, 2016

This notebook downloaded from http://mathworld.wolfram.com/notebooks/CellularAutomata/Rule62.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Rule62.html.

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

Initialization

rule=62;

Binary

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

Plot

V5

ElementaryCADiagram[rule]

V6

ElementaryCADiagram[rule]

V11

RulePlot[CellularAutomaton[62]]

RulePlot[CellularAutomaton[62],{{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]&/@ElementaryCATriangle[rule,10]

FindSequenceFunction[%,n]

Transpose[Rest[Map[FromDigits[#,2]&,DeleteCases[MapIndexed[Partition[#1,#2[[1]],#2[[1]],{1,1},{{}}]&,ElementaryCATriangle[rule,10]],{},{3}],{2}]]]

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

not found; construct as hybrid of two halves

First Part

(First[Partition[#,Floor[Length[#]/2]]]&/@Rest[ElementaryCATriangle[rule,10]])

FromDigits[#,2]&/@(First[Partition[#,Floor[Length[#]/2]]]&/@Rest[ElementaryCATriangle[rule,10]])

FindSequenceFunction[{1,3,6,13,27,54,109,219,438,877},n]//FullSimplify

RSolve[{a[n]==a[n-3]+3 2^(n-3),a[0]=0,a[1]=1,a[2]=3},a[n],n]

a[1]=0;
a[2]=1;
a[3]=3;
a[n_]:=a[n-3]+3 2^(n-3)

a/@Range[10]

Series[x(1+x)/((1-x)(1-2x)(1+x+x^2)),{x,0,10}]

<<DiscreteMath`RSolve`

FullSimplify[SeriesTerm[x(1+x)/((1-x)(1-2x)(1+x+x^2)),{x,0,n}],n∈Integers&&n>2]

l=Table[((4 (-7+9 (2^n)) (-I+(Sqrt[3]))-2 (-5 I+(Sqrt[3])) (E^((2 I n π)/3))-2 (I+3 (Sqrt[3])) (E^((4 I n π)/3)))/(42 (-I+(Sqrt[3])))),{n,0,10}]//Simplify

IntegerDigits[#,2]&/@l

l=IntegerDigits[#,2]&/@(Table[((4 (-7+9 (2^n)) (-I+(Sqrt[3]))-2 (-5 I+(Sqrt[3])) (E^((2 I n π)/3))-2 (I+3 (Sqrt[3])) (E^((4 I n π)/3)))/(42 (-I+(Sqrt[3])))),{n,0,10}]//Simplify)

ArrayPlot[PadLeft[#,10]&/@l]

Second Part

ArrayPlot[ElementaryCellularAutomaton[rule,40],Mesh->True]

l=MapIndexed[Take[#1,1-#2[[1]]]&,ElementaryCATriangle[rule,40]];

FromDigits[#,2]&/@Rest[l]

FindSequenceFunction[{1,1,7,1,23,57,15,177,471,121,1423,3761,983,11385,30095,7857,91095,240761,62863,728753,1926103,502905,5830031,15408817,4023255,46640249,123270543,32186033,373122007,986164345,257488271,2984976049,7889314775,2059906169,23879808399,63114518193,16479249367,191038467193,504916145551,131833994929},n]//FullSimplify

Sum[(2^(12k))+(2^(12k+3))+(2^(12k+4))+(2^(12k+5))+(2^(12k+6))+(2^(12k+10))+(2^(12k+11)),{k,0,(n-0)/2}]//Simplify

Table[%,{n,20}]

Sum[(2^(12k))+(2^(12k+1))+(2^(12k+2))+(2^(12k+3))+(2^(12k+7))+(2^(12k+8))+(2^(12k+10)),{k,0,n/2}]//Simplify

Table[%,{n,20}]

Sum[(2^(12k))+(2^(12k+4))+(2^(12k+5))+(2^(12k+7))+(2^(12k+9))+(2^(12k+10))+(2^(12k+11)),{k,0,n/2}]//Simplify

Table[%,{n,20}]