Integer sequence explained
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
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.
Computable and definable sequences
An integer sequence is computable if there exists an algorithm that, given
, calculates
, for all
. 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
is a
transitive model of
ZFC set theory. The transitivity of
implies that the integers and integer sequences inside
are actually integers and sequences of integers. An integer sequence is a
definable sequence relative to
if there exists some formula
in the language of set theory, with one free variable and no parameters, which is true in
for that integer sequence and false in
for all other integer sequences. In each such
, there are definable integer sequences that are not computable, such as sequences that encode the
Turing jumps of computable sets.
For some transitive models
of ZFC, every sequence of integers in
is definable relative to
; for others, only some integer sequences are. There is no systematic way to define in
itself the set of sequences definable relative to
and that set may not even exist in some such
. Similarly, the map from the set of formulas that define integer sequences in
to the integer sequences they define is not definable in
and may not exist in
. 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
contains all integer sequences, then the set of integer sequences definable in
will exist in
and be countable and countable in
.
Complete 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.
Examples
Integer sequences that have their own name include:
See also
References
External links
