, 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 which have received their own name include:
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
) 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.