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Wikipedia / Integer sequence

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, for the nth term: an explicit definition.

Alternatively, an integer sequence may be defined by a property which members of the sequence possess and other integers do not possess. For example, we can determine whether a given integer is a perfect number, , even though we do not have a formula for the nth perfect number.

Computable and definable sequences

An integer sequence is computable. The set of computable integer sequences is countable. The set of all integer sequences is uncountable (with cardinality equal to that of the continuum), and so not all integer sequences are computable.

Although some integer sequences have definitions, there is no systematic way to define what it means for an integer sequence to be definable in the universe or in any absolute (model independent) sense.

is a transitive model of ZFC set theory are actually integers and sequences of integers. An integer sequence is a definable, there are definable integer sequences that are not computable, such as sequences that encode the Turing jumps of computable sets.

. However, in any model that does possess such a definability map, some integer sequences in the model will not be definable relative to the model.

.

Complete sequences

A sequence of positive integers is called a complete sequence if every positive integer can be expressed as a sum of values in the sequence, using each value at most once.

Examples

Integer sequences that have their own name include:

See also

References

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