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# 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, … is formed according to the formula n2 − 1 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.

Examples

Integer sequences which have received their own name include:

Abundant numbers

Baum–Sweet sequence

Bell numbers

Binomial coefficients

Carmichael numbers

Catalan numbers

Composite numbers

Deficient numbers

Euler numbers

Even and odd numbers

Factorial numbers

Fibonacci numbers

Fibonacci word

Figurate numbers

Golomb sequence

Happy numbers

Highly totient numbers

Highly composite numbers

Home primes

Hyperperfect numbers

Juggler sequence

Kolakoski sequence

Lucky numbers

Lucas numbers

Padovan numbers

Partition numbers

Perfect numbers

Pseudoperfect numbers

Prime numbers

Pseudoprime numbers

Regular paperfolding sequence

Rudin–Shapiro sequence

Semiperfect numbers

Semiprime numbers

Superperfect numbers

Thue-Morse sequence

Ulam numbers

Weird numbers

Computable and definable sequences

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 (in other words, some sequences are definable but not computable). The set of all integer sequences is uncountable (with cardinality equal to that of the continuum); thus, almost all integer sequences are uncomputable and cannot be defined.

Complete sequences

An integer sequence 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.

See also

On-Line Encyclopedia of Integer Sequences

List of OEIS sequences

External links

Journal of Integer Sequences. Articles are freely available online.

Inductive Inference of Integer Sequences

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