Explicit form of recurrence relation when the recurring term is an exponent

Question

Suppose I have the sequence

$u_0 = 1$
$u_1 = 3$
$u_2 = 27$
$u_n = 3^{u_{n-1}}$, for integer $n > 0$

As you can see, the nth term is 3 raised to the power of $u_{n-1}$th term, which forms a sort of recurrence relation.

Is there a way to find an explicit form for this sequence? How?

Answer 1

Sometimes, this is written $u_n=\ ^n3$. This operation is called tetration.