I have been fascinated by integer sequences such as the look and say sequence and its many variant.
USUALLY such sequences grow like $ a n^b $ or like $ c n ^ d $ where $a,b,c$ are algebraic Numbers or algebraic number raised to an algrbraic power ( like $ \sqrt 2 ^{\sqrt3} $).
Very often we meet the golden mean or conways look and say constant ( = algebraic integer ).
For Some reason i was not able to find links to most of them.
But here are some ( i might add more later )
[1]
https://en.m.wikipedia.org/wiki/Kolakoski_sequence
[2]
https://en.m.wikipedia.org/wiki/Look-and-say_sequence
[3]
https://en.m.wikipedia.org/wiki/Golomb_sequence
[3] grows like $ \phi^{2 - \phi} n^{\phi -1} $.
All these closed form asymptotics are fascinating me.
Also this one ( no link or name ?? )
Consider the sequence
$1,2,2,3,3,4,4,4,...$
where $a_1=1,a_{n+1}\in\{a_n,a_n+1\}$, and $a_n$ is the number of times $n$ occurs in the sequence. Then if we assume that $a_n$ grows asymptotically as $\alpha n^\beta$, we get
$\alpha=\phi^{1/{\phi^2}}$
$\beta=1/\phi$.
For all $n$ the asymptotic expression is well within one unit of the actual $a_n$.
Etc etc
But recently I was thinking more of a fibonacci kind of look n say.
Let $f(1) = 1$.
$f(2) = f(2 - f(1)) = 1$.
$f(3) = f(3 - f(2)) + f(3 - f(1)) $
Etc
I found it here !!
So what is the limit of $ f(n) / f(n-1) $ when $n$ goes to positive $\infty$ ?
Does the limit have a closed form ?