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An integer sequence may be specified ''explicitly'' by giving a formula for its ''n''th 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 ''n''2 − 1 for the ''n''th 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 ''a''''n'', 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 EXTERNAL LINKS
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