[seqfan] Look and say -- my parity

[Eric Angelini]

Hello SeqFans,
we want the integers of W to 
describe the quantity of even and 
odd digits used so far in the sequebce
(including themselves).
We will use the "look and say" 
method to do that -- with the 
convention that "1" stands for odd
and "2" for even.

W starts with 22 because there are
indeed 2 even digits so far (two "2").
The next term will be 42 because
we have now altogether 4 even digits
in W (four "2"):

W=22,42,...

W is always extended with the smallest 
available  integer not leading to a 
contradiction. I guess we have:

W=22,42,62,82,3192,...

This last integer is not to be read
< 319 even digits so far >, of course,
but < 3 odd + 9 even digits so far >.

[The integer 9231 is available too -- but
we prefer 3192 as we "extend W with
the smallest available integer".
Please note that 3192 and 9231 both
beat 10231 that would fit also.]

I guess W will stop at some point, but
I don't know. This was computed by hand:

W=22,42,62,82,3192,11261,101132,141152,172181,
201212,221252,241292,271322,311342,351362,382391,411422,421462,...

The last integer must thus be read as: "We have used so far 42 odd digits
and 46 even digits to write W, including
this integer".
------------------------------------------
The convention < "1" represents an 
odd digit and "2"  an even one > is 
fun to explore together with the "look 
and say" methods (A or B).

For instance (method A, read from left 
to right, one chunk of equal digits after
the other):

1
11 ("one odd digit, on the above line")
21 ("two odd digits, on the above line")
1211 ("one even digit, then one odd digit on the above line")
111221 (1 odd, 1 even, 2 odd)
312211 (3 odd, 2 even, 1 odd)
212221 (2 odd, 2 even, 2 odd)
... etc.

Starting with 0 would lead to a fixed point:

0
12
1112
3112
3112...

Another fixed point is the well-known 22, of course. Are there more?

Best,
É.