Given the sequence $6,15,18,19,...,$ what is the recursive/explicit formula?

This is clearly not an arithmetic or geometric series but it does seem to satisfy:

\begin{align\*}
u(1) &= 6 \\
u(n) &= u(n-1) + 9^{1/n}
\end{align\*}

A general question: Given only a few terms I would guess that my solution is not unique. Is there any way to prove this other than by counter-example?

Are there any good strategies for determining these formulae other than intuition and trial&error?

Thank you,
Chris