Consider the sequence A126022 at OEIS
$$ 1,2,4,7,10,13,17,21,25,30, ... $$
We start with $a_1 = a(1) = 1$.
$a_n = a(n)$ and $^{-1}$ means functional inverse.
By $\lfloor a^{-1}(n+1) \rfloor$ we mean $\max K $ such that $a_K = a(K) $ and $a_K = n+1 $ or $a_K < n+1$.
( check the link If you are confused , Leroy explains it more clearly )
If this self-reference reminds you of the look and say sequence , that is normal.
In fact this question was inspired by a similar question I asked
And Also my mentor.
So here again I ask for understanding and more specific asymptotics with closed form values.
It might be intresting to note the very similar - in fact a motivation also for this question - equation pointed out by my mentor :
$ F'(x) = F^{-1}(x) $
That can be solved by plugging in
$$ F(x) = r x^t $$
And solving for $r,t$.
So I assume a good asymptotic to $a(n)$ is $ j x^k $
And of course the idea that $t = k$ and/or $r = j$ comes naturally.
So far the main ideas.
I have not done numerical tests yet.
It seems intuitive and easy , but I have No results at the moment.
A second question is the analogue sequence associated with $ G ' (x) = G^{-1}(x-1) $.