Continuum of the up-arrow notation halfways at higher level

Question

I found out that in the up-arrow notation, at higher levels of the arrow notation, especially with more than four arrows (hexation), in which I attempt to find the perfect continuum for $a \uparrow^{n} (k+0.5)$ for any integer $k$ (half fractions) for all $n>2$, where results are expected to be around $a \uparrow^{n-1} 3=a \uparrow^{n-2} a \uparrow^{n-2} a$; as well as $a \uparrow^{k+0.5} a$ for any integer $k$, which are expected to be closer to $a \uparrow^{k+1.5} 3$ rather than $a \uparrow^{k+1.5} 4$, for $k \ge 2$. This because it makes a much smoother curve for larger numbers using up-arrow notation with so many arrows. For example, $10 \uparrow^6 1.5$ tends to be closer to $ 10 \uparrow^5 3 = 10 \uparrow^4 10 \uparrow^4 10$, rather than $10 \uparrow^5 4 = 10 \uparrow^4 10 \uparrow^4 10 \uparrow^4 10$; and $10 \uparrow^{10} 1.5$ is really close to $10 \uparrow^9 3$. Also, $10 \uparrow^5 2.5$ is closer to $10 \uparrow^4 10 \uparrow^4 3$, and, for non-integer number of arrows, $10 \uparrow^{7.5} 10$ is closer to $10 \uparrow^8 3$ too!