Catalan’s Constant Continued Fraction

Author

Eric W. Weisstein
January 6, 2019

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

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

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

Terms

seq=ContinuedFraction[Catalan,256]

Plots

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

ContinuedFraction[Catalan,100]

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["CatalanConstantContinuedFractionBinaryPlot.gif",cf]

FindSequenceFunction

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

LinearRecurrence

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

GeneratingFunction

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

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

Compute

(10^9)

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

N[cons]

Precision[cons]

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

{ByteCount[cf],MaxMemoryUsed[]}/10^9. GB

DumpSave["/Developer/eww/ContinuedFractions/Catalan_970278792.mx",cf];

5×(10^9)

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

{N[cons],Precision[cons]}

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

{ByteCount[cf],MaxMemoryUsed[]}/10^9. GB

DumpSave["/Developer/eww/ContinuedFractions/Catalan_4851389025.mx",cf];//Timing//AbsoluteTiming

Read from .mx

Get["/Developer/eww/ContinuedFractions/Catalan_4851389025.mx"];//Timing//AbsoluteTiming

{Length[cf],ByteCount[cf]/10^9. GB}

High Water Marks

A099789/A099790

Length[cf]

(hwm=HighWaterMarks[cf]);//Timing

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

Length/@hwm

First Occurrences

A196461

Length[cf]

fo=FirstOccurrences[cf];//Timing

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

Take[fo,20]-1

BFile["~/b196461.txt",Take[fo,p[[1]]-1]-1]

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

n-Digit Terms

Length[cf]

digs=IntegerLength[cf];//Timing

(max=Max[digs])//Timing

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

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