Obtaining a recurrence from a rational generating function
Suppose that
$$ G(x) = \sum_{n=0}^\infty g_n x^n. $$
Using the given equation (ending at $i_m$, a better choice of variables), we have
$$
1 = (1-x^{i_1}-\cdots-x^{i_m}) G(x) =
\sum_{n=0}^\infty (1-x^{i_1}-\cdots-x^{i_m}) g_n x^n = \\
\sum_{n=0}^\infty g_n x^n - \sum_{n=i_1}^\infty g_{n-i_1} x^n - \cdots - \sum_{n=i_m}^\infty g_{n-i_m} x^n = \\
\sum_{n=i_m}^\infty (g_n - g_{n-i_1} - \cdots - g_{n-i_m}) x^n - H(x),
$$
where $\deg H(x) &lt; i_m$. Comparing coefficients, we find out that for $n \geq i_m$,
$$ g_n = g_{n-i_1} + \cdots + g_{n-i_m}. $$

More generally, you could have allowed coefficients in front of the monomials.