several messages (Fibonacci)

[Michele Dondi]

On Sun, 11 Dec 2005, kohmoto wrote:

>    %N A000001 a(n)=Prime(InversePrime(a(n-2))+a(n-1))

On Sun, 11 Dec 2005, kohmoto wrote:

>    %N A000001 a(n)=Fibonacci(InverseFibonacci(a(n-2))+a(n-1))

Both your sequences are of the form

   (1)    a(n)=F(G(a(n-2))+a(n-1)),

where G=F^{-1} N->N (or Z->Z) may not be well defined for all integers, 
but that doesn't matter, for a(n-2) is in Im(F) anyway.

Whatever, what you get is in any case a subsequence a(n)=F(b(n)) of
F(n), whereas the sequence b(n) of subscripts may be a more interesting 
object. Obviously it obeys the recursion

   (2)   b(n)=b(n-2)+F(b(n-1)),

so you don't mess with inverse functions. Also, while we're there we may 
also take into account

   (3)   b(n)=F(b(n-2))+F(b(n-1))

and

   (4)   b(n)=F(b(n-2))+b(n-1).

However I still consider this kind of sequences fundamentally 
uninteresting, except possibly for particular choices of F that would lead 
to some interesting arithmetic property.

If F(n) is pi(n), then the rate of growth of them will be slower 
than that of Fibonacci numbers (especially for (4)), but how much slower? 
How 'bout other asymptotic properties? A priori I don't exepect stonishing 
arithmetic properties though...

If F(n) is the n-th Fibonacci number, then (some of) said sequences _may_ 
be of some interest because the recursion formulas have a very similar 
form to that defining F(n), and while F(n) grows exponentially for large 
n, F(n) <= n for 0 <= n <= 5, so these sequences will be smaller than F(n) 
for some initial terms to boost up from some point on. Well, depending on 
the initial values, that is.


Just a few random thoughts,
Michele
-- 
Ain't got no future or a family tree,
But I know what a prince and lover ought to be,
I know what a prince and lover ought to be...
- Spin Doctors, "Two Princes"