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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