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Recurrence relation

Wikipedia

In mathematics, a recurrence relation is an equation that expresses the nth term of a sequence as a function of the k preceding terms, for some fixed k, which is called the order of the relation. Once k initial terms of a sequence are given, the recurrence relation allows computing recursively all the remaining terms of the sequence. Most general results on recurrence relations are about linear recurrences, which are recurrence relations such that the nth term is linear with respect to its preceding terms. Among them, linear recurrences with constant coefficients, and linear recurrences with polynomial coefficients are especially important. In the first case, this is because one can express the general term of the sequence as a closed-form expression of the index of the term. In the second case, this is because many common elementary and special functions have a Taylor series whose coefficients satisfy such a recurrence relation.

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