$u_{n+1} = a u_n +b u_{n-1} +c$ Is there a closed form formula for the general term of a recurrence sequence satisfying $$u_{n+1} = a u_n +b u_{n-1} +c$$               The sequence I am interested in satisfies $u_{n+1} = 8 u_n - 3 u_{n-1}  -4$.

This is a linear difference equation so can be stated as
$$
u = u_h + u_p
$$
with
$$
u_h(n+1)-au_h(n)-bu_h(n-1) = 0\\
u_p(n+1)-au_p(n)-bu_p(n-1) = c\\
$$
now making $u_h(n) = \phi^n$ and substituting we have
$$
\phi^n\left(\phi-a-b\phi^{-1}\right) = 0
$$
so
$$
\phi = \frac{1}{2} \left(a\pm\sqrt{a^2+4 b}\right)
$$
and
$$
u_h(n) = \frac{C_1}{2^n} \left(a-\sqrt{a^2+4 b}\right)^n+\frac{C_2}{2^n} \left(a+\sqrt{a^2+4 b}\right)^n
$$
and also
$$
u_p(n) = \frac{c}{1-a-b}
$$
so finally
$$
u(n) =  \frac{C_1}{2^n} \left(a-\sqrt{a^2+4 b}\right)^n+\frac{C_2}{2^n} \left(a+\sqrt{a^2+4 b}\right)^n + \frac{c}{1-a-b}
$$
