In the book Concrete Mathematics, chapter 2, section 2.2 -- sums and recurrences, page 26 (2nd edition), the authors talk about the following example:
Given the general recurrence
$ R(0) = \alpha $ $ R(n) = R(n-1) + \beta + \epsilon n $
The authors generalize the recurrence relation to:
$ R(n) = A(n)\alpha + B(n)\beta + C(n)\epsilon $
Employing the Repertoire Method, the authors plug in simple functions of $n$ in order to determine $A(n), B(n), C(n)$. So they discover:
Setting $R(n) = 1$ implies $\alpha = 1, \beta = 0, \epsilon = 0 \implies A(n) = 1$.
Setting $R(n) = n$ implies $\alpha = 0, \beta = 1, \epsilon = 0 \implies B(n) = n$.
Setting $R(n) = n^2$ implies $\alpha = 0, \beta = -1, \epsilon = 0 \implies C(n) = \frac{n^2 + n}{2}$.
Values for the first couple of terms of the recurrence:
$ \begin{eqnarray*} R(0) &=& \alpha \\ R(1) &=& \alpha + \beta + \epsilon \\ R(2) &=& \alpha + 2\beta + 3\epsilon \\ R(3) &=& \alpha + 3\beta + 6\epsilon \\ R(4) &=& \alpha + 4\beta + 10\epsilon \\ R(5) &=& \alpha + 5\beta + 15\epsilon \end{eqnarray*} $
I do not understand what is the process through which the values for $\alpha$, $\beta$, and $\epsilon$ are implied. I would like some help with that. Where exactly do we look and what do we math them against to see what they have to be?