<s>[INST] Recurrence equation: $u_n = 4u_{n−1} + 4u_{n−2}$ ; is $4x+4 = 4$ the characteristic equation? Given this recurrence equation:
$u_1 = 0, u_2 = 1$
$u_n = 4u_{n−1} + 4u_{n−2}$
Is the correct characteristic equation:
$4x+4 = 4$
EDIT:
Complete solve:
The characteristic equation is
$x^2-4x-4=0$
We solve the quadratic equation...
$\alpha = 5$ 
$\beta=-1$
So:
$u_n = c_1 \alpha^n + c_2 \beta^n$
We solve the equation...
$c_1 = 1/30$
$c_2 = 1/6$
Finally:
$u_n = \dfrac{5^n}{30} + \dfrac{(-1)^n}{6}$
 [/INST] Give Wilf's techniques a try... define $U(z) = \sum_{n \ge 0} u_{n + 1} u^z$,
and write the recurrence as:
$$
u_{n + 2} = 4 u_{n + 1} + 4 u_n \qquad u_1 = 0, u_2 = 1
$$
By properties of generating functions:
$$
\frac{U(z) - u_1 - u_2 z}{z^2} = 4 \cdot \frac{U(z) - u_1}{z} + 4 \cdot U(z)
$$
Maxima solves this as:
$$
U(z) = \frac{z}{1 - 4 z - 4 z^2}
     = \frac{1}{2^{5/2}} \cdot \frac{1}{1 - (2^{3/2} + 2) z}
         - \frac{1}{2^{5/2}} \cdot \frac{1}{1 + (2^{3/2} - 2) z}
$$
Two geometric series:
$$
u_{n + 1} = \frac{1}{2^{5/2}} \cdot
              \left( 
                 (2^{3/2} - 2)^n - (-1)^n (2^{3/2} + 2)^n
              \right)
$$
</s>