Tetration

Example

Let $a > 0$ and let $\exp_a(x) = a^x$ be real exponentiation to base $a$. For integers $n \ge 1$, the tetration ${}^{n}a$ is defined by $${}^{1}a = a, \qquad {}^{n+1}a = a^{\left({}^{n}a\right)}.$$ So ${}^{n}a$ is a tower of $n$ copies of $a$, evaluated from the top down. With ${}^{0}a = 1$, one has ${}^{n}a = \exp_a^{\,n}(1)$, the $n$-th iterate of $\exp_a$ applied to $1$.

Check for $a = 2$: ${}^{1}2 = 2$, ${}^{2}2 = 2^2 = 4$, ${}^{3}2 = 2^4 = 16$ and ${}^{4}2 = 2^{16} = 65536$. And $\exp_2(1) = 2$, $\exp_2(2) = 4$, $\exp_2(4) = 16$, $\exp_2(16) = 65536$, as claimed.