Summary
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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