"Que Sera, Sera" sequence.

[Richard Guy]

For expression of large numbers, see
The Book of Numbers (JHC & RKG, Springer,1996)
pp.59--62, where you can use Donald Knuth's
`uparrow' notation for the Ackerman numbers,
or Conway's `chained arrow' notation for even
bigger ones.    R.

On Tue, 23 Aug 2005, Alexandre Wajnberg wrote:

> 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