Definition:Recursive Sequence

Definition
A recursive sequence is a sequence where each term is defined from earlier terms in the sequence.

A famous example of a recursive sequence is the Fibonacci sequence:
 * $F_n = F_{n-1} + F_{n-2}$

The equation which defines this sequence is called a recurrence relation or difference equation.

In order for a recursive sequence to be defined, it is necessary to have the initial term (or terms) defined explicitly.

For example, in the above Fibonacci sequence, these are defined as:
 * $F_0 = 0, F_1 = 1$

Linear Recurrence Relation
A linear recurrence relation has the form:


 * $a_n y_{n+k} + a_{n-1}y_{n+k-1} + \cdots + a_0 y_k = b(k)$

where $a_0,\ldots,a_n$ are constants.

If $b(k) = 0$ the equation is homogeneous, otherwise it is inhomogeneous.