Khinchin's Constant Continued Fraction

Author

Eric W. Weisstein
January 6, 2019

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Constants/KhinchinsConstantContinuedFraction.nb.

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

©2019 Wolfram Research, Inc. except for portions noted otherwise

Terms

seq=ContinuedFraction[Khinchin,256]

FindSequenceFunction

FindSequenceFunction[Rest@seq,n]//Head//Timing

LinearRecurrence

FindLinearRecurrence[Rest[seq]]//Head//Timing

GeneratingFunction

(gf=FindGeneratingFunction[seq,x])//Head//Timing

Plots

ListPlot[Take[seq,100],Filling->Axis,PlotRange->All]

ArrayPlot[Transpose@PadLeft@IntegerDigits[Take[seq,100],2],PixelConstrained->3,Frame->False]

cf=ArrayPlot[Transpose@PadLeft@IntegerDigits[seq,2],PixelConstrained->3,Frame->False]

SetDirectory["/Users/eww/Sites/mathworld.wolfram.com/htdocs/images/gifs"];

Export["KhinchinConstantContinuedFractionBinaryPlot.gif",cf]

Convergents

With[{n=30},FromContinuedFraction/@Reverse[NestList[Most,ContinuedFraction[khinchin,n],n-1]]]

Through[{Numerator,Denominator}[%]]

Compute

110,000

Precision[cons=MathConstant["Khinchin"]]

{$MachineName,$TopDirectory,DateString[],$Version}//Column

Length[cf=ContinuedFraction[cons]]//Timing

{ByteCount[cf],MaxMemoryUsed[]}/10^6.MB

Incrementally Largest Terms

terms: A054866

A054870

110,000

Length[cf]

{$MachineName,$TopDirectory,DateString[],$Version}//Column

Take[cf,10]

(hwm=HighWaterMarks[cf,Method->"LowMemory"])//Timing

{hwm[[1]],hwm[[2]]-1}

First Occurrences

A054781

110,000

{$Version,Date[],Length[cf],(first=FirstOccurrences[cf,10^6,Method->"LowMemory"]);//Timing,MaxMemoryUsed[]/10^6. MB}//Column

Flatten[Position[first,0,1,20]]

Take[first,1000]

Loop

ContinuedFraction[Khinchin,20]

fo=FirstOccurrences[cf];//Timing

p=Take[Position[fo,0],10]//Flatten

Take[fo,20]-1

Take[fo-1,110]

BFile["~/b224851.txt",Take[fo,p[[1]]-1]-1]

ListPlot[Take[fo,p[[1]]-1]-1,AxesLabel->{Subscript[a,n],n},PlotRange->All]

n-Digit Terms

110,000

Length[cf]

digs=IntegerLength[cf];//Timing

Max[digs]//Timing

positions=Table[Flatten[Position[digs,n,1,20]],{n,6}];//Timing

data=MapIndexed[{#2[[1]],{#,cf[[#]]}}&,positions]

Grid[{#[[1]],Column@#[[2]]}&/@data,Dividers->All,Alignment->Left]

Khinchin and Lévy Constant

KhinchinList[x_,n_:200]:=MapIndexed[#1^(1/#2[[1]])&,Rest@FoldList[Times,1,Rest[ContinuedFraction[Mod[x,1],n+1]]]]
LevyList[x_,n_:200]:=MapIndexed[#1^(1/#2[[1]])&,Denominator/@FromContinuedFraction/@Reverse[NestList[Drop[#,-1]&,Rest[ContinuedFraction[Mod[x,1],n+1]],n]]]

<<Utilities`Typesetting`

With[{k=5000,c=Khinchin},
g1=ListPlot[KhinchinList[c,k],Joined->True,PlotStyle->Red,AxesLabel->{n,((Subscript[a, 1])(Subscript[⋯a, n]))^InlineFraction[1,n]},
PlotLabel->"Khinchin's constant",
Epilog->{Blue,Dashing[{.02}],Line[{{0,#},{k,#}}]&[2.685452001]}];
g2=ListPlot[LevyList[c,k],Joined->True,PlotStyle->Red,AxesLabel->{n,(Subscript[q, n])^InlineFraction[1,n]},
PlotLabel->"Lévy constant",
Epilog->{Blue,Dashing[{.02}],Line[{{0,#},{k,#}}]&[Exp[Pi^2/12/Log[2]]]}
];
]//Timing

GraphicsRow[{g1,g2},Spacings->-50,ImageSize->400]