In mathematics, a **recurrence relation** is an equation that recursively defines a sequence, once one or more initial terms are given: each further term of the sequence is defined as a function of the preceding terms.

The term **difference equation** sometimes (and for the purposes of this article) refers to a specific type of recurrence relation. However, "difference equation" is frequently used to refer to *any* recurrence relation.

An example of a recurrence relation is the logistic map:

with a given constant *r*; given the initial term *x*_{0} each subsequent term is determined by this relation.

Some simply defined recurrence relations can have very complex (chaotic) behaviours, and they are a part of the field of mathematics known as nonlinear analysis.

Solving a recurrence relation means obtaining a closed-form solution: a non-recursive function of *n*.

Read more about Recurrence Relation: Fibonacci Numbers, Relationship To Differential Equations

### Famous quotes containing the words recurrence and/or relation:

“Forgetfulness is necessary to remembrance. Ideas are retained by renovation of that impression which time is always wearing away, and which new images are striving to obliterate. If useless thoughts could be expelled from the mind, all the valuable parts of our knowledge would more frequently recur, and every *recurrence* would reinstate them in their former place.”

—Samuel Johnson (1709–1784)

“The foregoing generations beheld God and nature face to face; we, through their eyes. Why should not we also enjoy an original *relation* to the universe? Why should not we have a poetry and philosophy of insight and not of tradition, and a religion by revelation to us, and not the history of theirs?”

—Ralph Waldo Emerson (1803–1882)