For which n does the inequality $2 \uparrow^{n+1}n > 3\uparrow^n 3 +2$ hold? For which n does the following inequality hold ?
$$2 \uparrow^{n+1}n > 3\uparrow^n 3 + 2$$
where $\uparrow$ stands for knuth's up-arrow notation.
I need this inequality to prove that
$$f_{\omega+1}(n) > G(n)$$
for $n\ge 8$
where $f_{\omega+1}(n)$ is a function from the
fast growing hierarchy and G(n) is Graham's 
sequence
$$G(1) = 3\uparrow^4 3$$
$$G(n+1) = 3\uparrow^{G(n)} 3$$
for all n > 0.

The inequality holds precisely when $n \ge 4$.
When $n=3$, we have 
$$2 \uparrow^4 3 = 2 \uparrow^3 4 = 2\uparrow^2 (2 \uparrow^2 4) = 2 \uparrow^2 65536 < 3\uparrow^3 3 + 2 = 3\uparrow^2(3\uparrow^2 3) +2 = 3\uparrow^2 7625597484987 + 2$$
Using the fact that $2 \uparrow^m (n+2) > 3\uparrow^m n + 2$,(proven here) we have for $n \ge 4$:
$$2 \uparrow^{n+1} n \ge 2\uparrow^{n+1} 4 = 2\uparrow^n(2\uparrow^n 4) > 2 \uparrow^n 5 > 3 \uparrow^n 3 + 2$$
