Rekurrensrelasjon

Rekurrensrelasjon, also known as a recurrence relation, is an equation that recursively defines a sequence where each term is defined as a function of the preceding terms. This means that to find a specific term in the sequence, you need to know the values of one or more of the previous terms. The simplest form of a recurrence relation is an explicit recurrence relation, where the next term depends directly on the previous term, often in a linear fashion. However, recurrence relations can also be more complex, involving multiple previous terms or even non-linear functions.

To fully define a sequence using a recurrence relation, initial conditions or base cases are necessary. These are the starting values of the sequence, typically the first one or few terms. Without these, the recurrence relation would generate an infinite number of possible sequences. Solving a recurrence relation involves finding a closed-form expression for the nth term of the sequence, which allows direct calculation of any term without needing to compute all the preceding ones. This is analogous to finding a general formula for an arithmetic or geometric sequence.

Recurrence relations are fundamental in various fields, including computer science, where they are used to analyze the time complexity of algorithms, and mathematics, particularly in combinatorics and discrete mathematics. They provide a powerful way to model and understand processes that evolve over time or stages.