Sequences containing all finite sequences

[Christian G.Bower]

> Which sequences contain all finite sequences of non-negative integers as
> subsequences?  (One can also add 1, and look for sequences containing just
all
> finite sequences of positive integers as subsequences.  However, the
natural
> examples I found all involved non-negative integers, so that's the way I'm
> framing the problem.)

...

> Are there other sequences with this property in the OEIS -
> or that should be in
> the OEIS?  Note that all 3 of these are tables, where every finite sequence
> occurs in a row in an obvious way.  It would be nice to find sequences not
> defined in this way.

If I take a "typical" irrational number and express it in binary, e.g.

ID Number: A004593
URL:       http://www.research.att.com/projects/OEIS?Anum=A004593
Sequence:  1,0,1,0,1,1,0,1,1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0,0,1,0,1,1,0,
           0,0,1,0,1,0,0,0,1,0,1,0,1,1,1,0,1,1,0,1,0,0,1,0,1,0,1,0,0,1,
           1,0,1,0,1,0,1,0,1,1,1,1,1,1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0,0,
           1,0,0,0,0,0,0,0,1
Name:      Expansion of e in base 2.

I would expect it to contain every finite binary sequence.  I suspect
this is very difficylt to prove though.  If it does contain every
binary sequence, then the sequence of run lengths:

1,1,1,1,2,1,6,4,4,1,1,1,1,1,3,1,...
Not in EIS

contains every finite sequence of positive integers, thus one might
expect A048821

ID Number: A048821
URL:       http://www.research.att.com/projects/OEIS?Anum=A048821
Sequence:  1,2,1,1,1,3,2,3,1,2,4,1,1,2,3,1,2,5,1,3,2,3,1,3,1,2,3,2,1,4,
           1,4,2,1,1,2,2,1,2,1,1,1,5,2,1,1,1,2,1,1,2,1,2,1,1,3,1,2,2,5,
           1,1,2,2,2,1,3,2,1,3,4,2,1,3,1,3,1,1,1,2,2,1,2,1,5,3,3,1,1,1,
           1,1,1,1,3,2,1,2,1,2,2,3
Name:      Lengths of runs in A048820.
References S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 441-443.
Links:     Steven Finch, Minkowski's Question Mark Function
Formula:   Defined by the property that .0^a 1^b 0^c 1^d ... = 1 / (a + 1/ (b
+ 1/ (c +
              ...))) as a continued fraction.

to contain every positive sequence.

ID Number: A048820
URL:       http://www.research.att.com/projects/OEIS?Anum=A048820
Sequence:  0,1,1,0,1,0,1,1,1,0,0,1,1,1,0,1,1,0,0,0,0,1,0,1,1,0,0,0,1,0,
           0,1,1,1,1,1,0,1,1,1,0,0,1,1,1,0,1,1,1,0,1,1,0,0,0,1,1,0,1,1,
           1,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,0,1,0,1,0,0,0,0,0,1,1,0,1,
           0,1,1,0,1,0,0,1,0,0,1,0
Name:      Binary expansion of fixed point of Farey function (or Minkowski's
              question mark function).

How about other sequences defined by irrational numbers like continued
fractions.  Obviously e does not have a typical cf expansion, but what
about pi?

Christian