<s>[INST] Finding the solution to this specific recurrence relation What would be the solution to $a_n = 7a_{n−2} + 6a_{n−3}$ with $a_0 = 9$,
$a_1 = 10$, and $a_2 = 32$
I can find it for a specific value of (n), but not for just a general solution.  Thanks!
 [/INST] Use generating functions. Define $A(z) = \sum_{n \ge 0} a_n z^n$, and write the recurrence as:
$$
a_{n + 3} = 7 a_{n + 1} + 6 a_n \quad a_0 = 9, a_1 = 10, a_2 = 32
$$
By properties of ordinary generating funtions:
$$
\frac{A(z) - a_0 - a_1 z - a_2 z^2}{z^3} = \frac{A(z) - a_0}{z} + 6 A(z)
$$
Writing as partial fractions:
$$
A(z) = 4 \cdot \frac{1}{1 - 3 z} 
        - 3 \cdot \frac{1}{1 + 2 z} 
        + 8 \cdot \frac{1}{1 + z}
$$
Expanding the various geometric series:
$$
a_n = 4 \cdot 3^n - 3 \cdot (-2)^n + 8 \cdot (-1)^n
$$
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