# Sequence

A sequence is a multiset that is an ordered list.

**AKA:**Series, Ordered Multiset, Ordered Sequence.**Context:**- It can have a Sequence Length.
- It can (typically) have:
- an Infium Element, (e.g. not a Left-Open Interval)
- an Supremum Element, (e.g. not a Right-Open Interval).

- It can range from:
- being an Empty Sequence (with zero Members) or a Non-Empty Sequence.
- to being a Degenerate Sequence, with One Member.
- to being a Finite Sequence (string).
- to being an Infinite Sequence.

- It can be in a Subsequence Relation (be a subsequence/segment to) another Sequence.
- It can be the Function Input to an Sequence Input Function.
- It can be represented by a Sequence Data Structure.
- It can be denoted as: (a1, a2, ...) or {a1 < a2 < ...}.

**Example(s):**- a Symbol String, such as a Tuple or a Vector.
- a Word Sequence.
- a Number Sequence, such the Arithmetic Sequence: (1, 1, 2, 3, ...).
- a Sequential Dataset.
- a Tuple Sequence.
- (a, z, c).
- {c ≤ e ≤ e ≤ l}, an Ordered Set.
- {3 ≤ 3 ≤ 4 ≤ 5}, a Numeric Sequence.
- {0 < 1 < 2 < 3... < ∞}, The N0 Natural Number Sequence.
- {-∞ ... < -1 ... < Zero ... < ⅛ ... < 1 ... < 3.14... <
*π*... < ∞} - a Natural Language Sentence, such as “
*I went home .*" - the Number Line.
- a Graph Edge Sequence.

**Counter-Example(s):****See:**Unordered Relation, Permutation.

## References

### 2013

- (Wikipedia, 2013) ⇒ http://en.wikipedia.org/wiki/Sequence
- In mathematics, informally speaking, a
**sequence**is an ordered list of objects (or events). Like a set, it contains members (also called*elements*, or*terms*). The number of ordered elements (possibly infinite) is called the*length*of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. Most precisely, a sequence can be defined as a function whose domain is a countable totally ordered set, such as the natural numbers.For example, {M, A, R, Y} is a sequence of letters with the letter 'M' first and 'Y' last. This sequence differs from {A, R, M, Y}. Also, the sequence {1, 1, 2, 3, 5, 8}, which contains the number 1 at two different positions, is a valid sequence. Sequences can be

*finite*, as in this example, or*infinite*, such as the sequence of all even positive integers {2, 4, 6,...}. Finite sequences are sometimes known as*strings*or*words*and infinite sequences as*streams*. The empty sequence { } is included in most notions of sequence, but may be excluded depending on the context.

- In mathematics, informally speaking, a

- http://en.wikipedia.org/wiki/Sequence#Examples_and_notation
- A sequence can be thought of as a list of elements with a particular order. Sequences are useful in a number of mathematical disciplines for studying functions, spaces, and other mathematical structures using the convergence properties of sequences. In particular, sequences are the basis for series, which are important in differential equations and analysis. Sequences are also of interest in their own right and can be studied as patterns or puzzles, such as in the study of prime numbers.
There are a number of ways to denote a sequence, some of which are more useful for specific types of sequences. One way to specify a sequence is to list the elements. For example, the first four odd numbers form the sequence {1,3,5,7}. This notation can be used for infinite sequences as well. For instance, the infinite sequence of positive odd integers can be written {1,3,5,7,...}. Listing is most useful for infinite sequences with a pattern that can be easily discerned from the first few elements. Other ways to denote a sequence are discussed after the examples.

- A sequence can be thought of as a list of elements with a particular order. Sequences are useful in a number of mathematical disciplines for studying functions, spaces, and other mathematical structures using the convergence properties of sequences. In particular, sequences are the basis for series, which are important in differential equations and analysis. Sequences are also of interest in their own right and can be studied as patterns or puzzles, such as in the study of prime numbers.

### 2009

- WordNet
- serial arrangement in which things follow in logical order or a recurrent pattern; "the sequence of names was alphabetical"; "he invented a ...
- a following of one thing after another in time; "the doctor saw a sequence of patients"
- film consisting of a succession of related shots that develop a given subject in a movie
- arrange in a sequence
- succession: the action of following in order; "he played the trumps in sequence"
- determine the order of constituents in; "They sequenced the human genome"
- several repetitions of a melodic phrase in different keys

- Wiktionary en.wiktionary.org/wiki/sequence
- A set of things next to each other in a set order; a series; A series of musical phrases where a theme or melody is repeated, with some change ...

- http://www.isi.edu/~hobbs/bgt-sequences.text
- A sequence of length n is a function whose domain is the first n positive integers.
`(11)`

```
(forall (s)
(iff (sequence s)
(exists (s1 s2 n)
(and (function0 s s1 s2)(ints s1 1 n))))))
```