# Applied Combinatorics

## Chapter9Recurrence Equations

We have already seen many examples of recurrence in the definitions of combinatorial functions and expressions. The development of number systems in Appendix B lays the groundwork for recurrence in mathematics. Other examples we have seen include the Collatz sequence of Example 1.8 and the binomial coefficients. In Chapter 3, we also saw how recurrences could arise when enumerating strings with certain restrictions, but we didn’t discuss how we might get from a recursive definition of a function to an explicit definition depending only on $$n\text{,}$$ rather than earlier values of the function. In this chapter, we present a more systematic treatment of recurrence with the end goal of finding closed form expressions for functions defined recursively—whenever possible. We will focus on the case of linear recurrence equations. At the end of the chapter, we will also revisit some of what we learned in Chapter 8 to see how generating functions can also be used to solve recurrences.