Bowers array notation : $f_{\omega^\omega}(n)\approx [n,...,n]$ ($n$ times)

Question

I learnt at this site that

$$\large f_{\omega^\omega}(n)\approx \underbrace{[n,...,n]}_{n\ n's}$$

For a simular approximation

$$\large f_{\omega^2}(n)\approx \underbrace{n\rightarrow n\rightarrow...\rightarrow n\rightarrow n}_{n\ n's}\ =:\ g(n)$$

Wythagoras concreted this approximation to the inequality

$$\large g(n+1)<f_{\omega^2}(n)<g(n+2)\ \ for\ \ n\ge 2$$

Is there a similar inequality in this case ?

Answer 1

It should be familiar to you that $f_{\omega}(3)\approx\{3,3,3\}=h(3)$. Similiarly, $f_{\omega^2}(4)\approx\{4,4,4,4\}=h(4)$, and in general $$h(n+1) << f_{\omega^\omega}(n) < h(n+2)$$

for $n \geq 3$. See also this answer.