I am trying to understand the following segment of my notes:

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> [![enter image description here](https://i.stack.imgur.com/7rD70.png)](https://i.stack.imgur.com/7rD70.png).
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I will try to explain what I think I understand and point out the parts I don't understand.

First, there is a result in my notes (not shown here) proving that a rational generating function defines a homogeneous recurrence relation. Then, I consult the following section of my notes:

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> [![enter image description here](https://i.stack.imgur.com/Zw2NS.png)](https://i.stack.imgur.com/Zw2NS.png).
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This is a procedure for turning a homogeneous recurrence relation into a generating function (in its rational form). Given the rational form $f(x) = \frac{1 + 7x}{1 + x - 6x^2}$, we see that the coefficients that define the homogeneous recurrence relation (i.e. the $b\_1, \dots, b\_k$) are located in the denominator, so we conclude that $a\_n = 6 a\_{n - 2} - a\_{n - 1}$.

**I don't understand** how the initial conditions $a\_0 = 1$ and $a\_1 = 6$ are determined through long division. I am only vaguely familiar with polynomial division so I may be missing something obvious, but isn't $\frac{1 + 7x}{1 + x - 6x^2}$ simply equal to $1 + 7x$, because the degree of the numerator is less than the degree of the denominator? How does this help me determine the initial conditions?

According to my notes (not shown here), the "term formula" as I call it for the $i$th coefficient $a\_i$ of the generating function can be found by factoring the denominator of $\frac{1 + 7x}{1 + x - 6x^2}$ into a suitable form, applying the method of partial fractions and using a result regarding the term formula for $\frac{1}{(1 - cx)^k}$ for $c \in \mathbb C$. Eventually I get that $a\_i = \frac{9}{5} \cdot 2^i - \frac{4}{5} (-3)^i$, which is the same result as the one in my notes, and from this we can determine the initial conditions $a\_0$ and $a\_1$.

I appreciate any help.