# 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 *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:

- Abundant numbers
- Bell numbers
- Binomial coefficients
- Carmichael numbers
- Catalan numbers
- Composite numbers
- Deficient numbers
- Euler numbers
- Even and odd numbers
- Factorial numbers
- Fibonacci numbers
- Figurate numbers
- Happy numbers
- Highly totient numbers
- Highly composite numbers
- Hyperperfect numbers
- Lucas numbers
- Perfect numbers
- Pseudoperfect numbers
- Prime numbers
- Pseudoprime numbers
- Semiperfect numbers
- Semiprime numbers
- Weird numbers

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

- Journal of Integer Sequences. Articles are freely available online.

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