"Que Sera, Sera" sequence.

[Alexandre Wajnberg]

Hello friends, Hello Éric,

I'm submitting a self-describing sequence which "exploded" in my hands  
and I need your help!

------------------------------
SEQUENCE:
1 2 33 444444444444444444444444444444444
55555555555555555555555555555555555555555555555...5555555555555555555555 
555555555555   666...666   777...777   888...888   999...999    
101010...101010   111111...111111   121212...121212
NAME:
"Que Sera, Sera" sequence: self describing sequence where a(n) gives  
the number of <number (n+1) > which will be concatenated to form  
a(n+1); starting with 1.
COMMENT :
This infinite sequence increases very rapidly: the order of magnitude  
of a(4) is 4,44*10^32, and of a(5) is 5,55*10^  
444444444444444444444444444444443.
If, in the definition, one replaces <number (n+1)> by <digit (n+1)>,  
the sequence is finite and ends at the ninth term (in base 10, first  
offset = 1).
EXEMPLES:
a(1) says: there will be one 2 in a(2).
a(2)=2 because a(1) said so; and a(2)=2 says: there will be two 3's in  
a(3).
a(3)=33 because a(2) said so; and also a(3) says: there will be thirty  
three 4's in a(4).
Therefore, a(4)= 444444444444444444444444444444444 (33 times the digit  
4).
And a(5)= 555555555555555...555 (thus 444444444444444444444444444444444  
times the digit 5, or approximately <4,44*10^32> times the digit 5).
a(9)=999...999 says: there will be 999...999 times the substring "10"  
in a(10).
a(10)= 101010...101010 because a(9) told so;  and a(10) says: there  
will be 101010...101010 times "11" in a(11).
%Y A000001 Cf. A001462, A001463, A103320, A102357, A076782
%O A000001 1,2
%K A000001 ,base,infinite,nonn,
%A A000001 Alexandre Wajnberg (alexandre.wajnberg at ulb.ac.be), Aug 22  
2005
------------------------------

Here, the seed is 1.
What is funny is that if the seed is 0, the sequence is really short!  
One term: a(1)=0.
[But if the seed is 2 or more, the growth is even more explosive.
Ex: 2  22  3333333333333333333333 (to compare to 33).]


My questions are:

1) How to express the terms coming after a(4)?
I think [to verify!]:
-a(5)= 5,55...*10^444444444444444444444444444444443 =  
5,55...*10^[(4,44...*10^32)-1]
-a(6) would be 6,66...*10^{5,55...*10^[(4,44...*10^32)-1]-1}
-a(7) would be 7,77...*10^[a(6)-1] =  
7,77...*10^[6,66...*10^{5,55...*10^[(4,44...*10^32)-1]-1}-1]
Etc...
And how to express their order of magnitude?
All these powers of powers of powers of ten make me feel giddy!

2) How to express those big numbers with more concise and clear  
expressions? (with googolplexes? numbers of Graham?...)? Some of you  
have the technical skill, isn't it?!  :-)

3) Keywords: on one hands, this seq is easy; on the other, terms next  
to a(4) seem to me becoming rather hard.

4) A more philosophical reflection. Usually, the "infiniteness" (or  
"infinityness") of a sequence is a "horizontally defined" one: the  
number of terms is infinite, but the value of each of them is finite  
(although they may be very big).
Here, it's something different. After a very small number of terms, the  
terms themselves are reaching so big values — not really infinite ones  
ok!, but here we are! It is precisely what I'm trying to figure out!  —  
that it seems to me that we could think to <<how to define another  
infinityness, a "vertically defined" one, for certain kind of  
sequences>>.

And more: is it thinkable to define sequences whose terms carry  
different kinds of infinity?


Best regards.

Alexandre

PS: http://www.webfitz.com/lyrics/Lyrics/1956/211956.html
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