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Concept
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Résumé
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.
Integer sequences that have 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 composite numbers
Highly totient numbers
Home primes
Hyperperfect numbers
Juggler sequence
Kolakoski sequence
Lucky numbers
Lucas numbers
Motzkin numbers
Natural numbers
Padovan numbers
Partition numbers
Perfect numbers
Prime numbers
Pseudoprime numbers
Recamán's sequence
Regular paperfolding sequence
Rudin–Shapiro sequence
Semiperfect numbers
Semiprime numbers
Superperfect numbers
Thue–Morse sequence
Ulam numbers
Weird numbers
Wolstenholme number
An integer sequence is a computable sequence if there exists an algorithm which, given n, calculates an, for all n > 0. 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.
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