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Enrique Zeleny
Universidad Autonoma de Puebla
Puebla, Pue.

Registered: Oct 2003
Posts: 4

Liouville's constant continued fraction

Searching some number that shows the large term phenomenon in his continued fraction representation, like in Champernowne constant (Sloane's A030167), I tried Liouville's constant and I found this phenomenon and that the expansion has an nested structure (Sloane and Weisstein don't say anything about this).
Here is the code:

Needs["Graphics`Graphics`"]

(*Take 3 hours in a 2 Ghz PC*)

liouville =
ContinuedFraction[
NSum[1/10^n!, {n, 1, Infinity}, WorkingPrecision -> 2000000], 500];

MapIndexed[List, (Length[IntegerDigits[#1]] & ) /@ liouville]

LogListPlot[%, PlotJoined -> True, AspectRatio -> 1/5,
PlotRange -> All];

the number of digits (9's) in the terms of the expansion are

Union[First /@ %%]

Table[(n - 1) n!, {n, 20}]

values 1, 8, 9, 10, 11, 99 are repeated, but in different order and at different lengths, terms with many 9's are represented with black cells (value 100) and a white cell means that you must see next row.

ArrayPlot[(PadRight[#1, 9] &) /@
Split[Take[liouville, {2, 296}] /. n_ /; n > 99 -> 100, #1 < 100 &] /.
Thread[{0, 1, 8, 9, 10, 11, 99, 100} -> {0, 2, 3, 4, 5, 6, 7, 10}]]



Greetings from Puebla, Mexico

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Old Post 01-30-2005 12:36 AM
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qplwer


Registered: Nov 2006
Posts: 1

reference

This continued fraction was discovered by Shallit:

Shallit, J. O.
Simple continued fractions for some irrational numbers. II.
J. Number Theory 14 (1982), no. 2, 228--231.

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Old Post 11-18-2006 01:19 AM
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