"Que Sera, Sera" sequence.

[hv at crypt.org]

Alexandre Wajnberg <alexandre.wajnberg at skynet.be> wrote:
[...]
:1) How to express the terms coming after a(4)?

For online information, I'd suggest the Wikipedia articles
(<http://en.wikipedia.org/>) on:
  tetration
  Knuth's up-arrow notation
  hyper operators
  Conway chained arrow notation
and the overview article:
  Large numbers

Using Knuth's up-arrow notation, for example, we have:
  10 = 10^^1
  x = 10^^y  =>  10^x = 10^^(y+1)
.. though I don't know how to derive an exact y for a given x.

I'd guess that a(4) is around 10^^1.1, in which case a(n) is around
10^^(n - 2.9 + f(n)) where (for suitable values of 2.9) I suspect
|f(n)| is O(k.slog_10(n)).

These notations can indicate magnitude, but I don't think they can tell
you much about actual value - I'm not aware of any such notation that
would help you consider things such as "which a(n) are divisible by 3?",
but would expect such questions to require proofs based on cycles or
other generic arguments rather than on absolute values.

Hugo