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Integer sequence

Ordered list of whole numbers

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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 (sequence A000045 in the OEIS). The sequence 0, 3, 8, 15, ... is formed according to the formula $n^{2}-1$ 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, (sequence A000396 in the OEIS), even though we do not have a formula for the nth perfect number.

01Computable and definable sequences

An integer sequence is computable if there exists an algorithm that, given $n$ , calculates $a_{n}$ , 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.

Suppose the set $M$ is a transitive model of ZFC set theory. The transitivity of $M$ implies that the integers and integer sequences inside $M$ are actually integers and sequences of integers. An integer sequence is a definable sequence relative to $M$ if there exists some formula $P(x)$ in the language of set theory, with one free variable and no parameters, which is true in $M$ for that integer sequence and false in $M$ for all other integer sequences. In each such $M$ , there are definable integer sequences that are not computable, such as sequences that encode the Turing jumps of computable sets.

For some transitive models $M$ of ZFC, every sequence of integers in $M$ is definable relative to $M$ ; for others, only some integer sequences are. There is no systematic way to define in $M$ itself the set of sequences definable relative to $M$ and that set may not even exist in some such $M$ . Similarly, the map from the set of formulas that define integer sequences in $M$ to the integer sequences they define is not definable in $M$ and may not exist in $M$ . However, in any model that does possess such a definability map, some integer sequences in the model will not be definable relative to the model.

If $M$ contains all integer sequences, then the set of integer sequences definable in $M$ will exist in $M$ and be countable and countable in $M$ .

02Complete sequences

A sequence of positive integers 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.

03Examples

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Sources and credits

This article is adapted from the Wikipedia article “Integer sequence”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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Goteborg ciag Fibonacciego.jpg by Andrzej Otrębski, CC BY-SA 4.0

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Integer sequence

01Computable and definable sequences

02Complete sequences

03Examples

Sources and credits

Related topics

Constant-recursive sequence

On-Line Encyclopedia of Integer Sequences

List of integer sequences