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Integer sequence

From Academic Kids

In mathematics, an integer sequence is a sequence (i.e., an ordered list) of integers.

An integer sequence may be specified explicitly by giving a formula for its nth term, or implicitly by giving a relationship between its terms. For example, the sequence 0, 1, 1, 2, 3, 5, 8, 13, ... (the Fibonacci sequence) is formed by starting with 0 and 1 and then adding any two consecutive terms to obtain the next one: an implicit description. The sequence 0, 3, 8, 15, ... is formed according to the formula n2 − 1 for the nth term: an explicit definition.

Integer sequences which have received their own name include:

An integer sequence is a computable sequence, if there exists an algorithm which given n, calculates an, for all n > 0. An integer sequence is a definable sequence, if there exists some statement P(x) which is true for that integer sequence x and false for all other integer sequences. The set of computable integer sequences and definable integer sequences are both countable, with the computable sequences a proper subset of the definable sequences. The set of all integer sequences is uncountable; thus, almost all integer sequences are uncomputable and cannot be defined.

See also

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Topics in mathematics related to quantity

Numbers | Natural numbers | Integers | Rational numbers | Constructible numbers | Algebraic numbers | Computable numbers | Real numbers | Complex numbers | Split-complex numbers | Bicomplex numbers | Hypercomplex numbers | Quaternions | Octonions | Sedenions | Superreal numbers | Hyperreal numbers | Surreal numbers | Nominal numbers | Ordinal numbers | Cardinal numbers | p-adic numbers | Integer sequences | Mathematical constants | Large numbers | Infinity
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