How does a recurrence relation determine later values in a sequence?

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combinatorics Figure 1: Ferrers' partitioning diagram for 14.

In combinatorics, a recurrence relation determines later values in a sequence by expressing a term as a function of one or more preceding terms.12

To calculate the sequence, the relation must be paired with a set of initial values, such as $f_{0}, f_{1}, \dots, f_{k - 1}$ . Once these starting values are established, the recurrence relation acts as a rule to generate subsequent terms iteratively. For example, in the relation $f_{n} = f_{n - 1} + f_{n - 2}$ , each new term is found by summing the two immediately preceding values. By applying this rule repeatedly starting from the known initial values, every subsequent term in the sequence can be determined.2