What is the pattern to this sequence?

In cases like this, it is recommended to try and visit the OEIS.

Of course, with such pattern recognition questions, you can let the next value be anything.

My suspicion is that $u_{n+1} = 4 u_{n} + (-1)^n$.

Using the recurrence $a_n = 4a_{n-2} +3a_{n-1}$, which this sequence appears to satisfy, you can derive an explicit formula for $a_n$, namely:

$a_n = \frac{4^n - (-1)^n}{5}$

It's exponential - it grows too fast to be quadratic. You can find a quadratic function that passes through any three of those points, but it won't pass through the others at the same time.