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recurrence relation (Definition)

A recurrence relation is an equation which gives the value of an element of a sequence in terms of the values of the sequence for smaller values of the position index and the position index itself. If the current position $ n$ of a sequence $ s$ is denoted by $ s_n$, then the next value of the sequence expressed as a recurrence relation would be of the form

$\displaystyle s_{n+1} = f(s_1,s_2,\ldots,s_{n-1},s_n,n) $

where $ f$ is any function. An example of a simple recurrence relation is

$\displaystyle s_{n+1} = s_n + (n+1) $

which is the recurrence relation for the sum of the integers from $ 1$ to $ n+1$. This could also be expressed as

$\displaystyle s_n = s_{n-1} + n $

keeping in mind that as long as we set the proper initial values of the sequence, the recurrence relation indices can have any constant amount added or subtracted.

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"recurrence relation" is owned by rspuzio. [ full author list (2) | owner history (2) ]
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See Also: Berlekamp-Massey algorithm, equation, finite difference

Other names:  difference equation

Cross-references: indices, integers, sum, function, current, index, terms, sequence, equation
There are 35 references to this entry.

This is version 4 of recurrence relation, born on 2001-11-04, modified 2005-11-27.
Object id is 664, canonical name is RecurrenceRelation.
Accessed 11605 times total.

AMS MSC11B37 (Number theory :: Sequences and sets :: Recurrences)
 03D20 (Mathematical logic and foundations :: Computability and recursion theory :: Recursive functions and relations, subrecursive hierarchies)

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