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# sequence

# Sequences

Given any set $X$, a *sequence* in $X$ is a function $f\colon\mathbb{N}\to X$ from the set of natural numbers to $X$. Sequences are usually written with subscript notation: $x_{0},x_{1},x_{2}\dots$, instead of $f(0),f(1),f(2)\dots$.

# Generalized sequences

One can generalize the above definition to any arbitrary ordinal. For any set $X$, a *generalized sequence* or *transfinite sequence* in $X$ is a function $f\colon\omega\to X$ where $\omega$ is any ordinal number. If $\omega$ is a finite ordinal, then we say the sequence is a *finite sequence*.

Defines:

generalized sequence, transfinite sequence, finite sequence

Related:

ConvergentSequence

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

03E10*no label found*40-00

*no label found*

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## Attached Articles

## Corrections

sequences synonym by akrowne ✓

naturals by vitriol ✘

Infinite/finite sequences by Henry ✓

synonym by CWoo ✓

naturals by vitriol ✘

Infinite/finite sequences by Henry ✓

synonym by CWoo ✓

## Comments

## What is sequence anyway

From one high school book from 1964, revisited by Slovene mathematician Ivan Vidav and written by Alojzij Vadnal goes this simple definition for sequence:

Sequence is any number set, which is arranged in a way that one number comes first, one second, one third and it is possible for every number of the set to define at which place of the sequence it stands.

Question is: function and number set can not be the same thing? Instead of functional notation f(0), f(1), f(2), ... we use {x_0, x_1, x_2, ... } what in the other side shows a structure of a set.

Am I missing some here? Once I have done one similar "ambiguity" when I said: if we *do this and that*, then we get a set of integer sequences. I should simply say: then we get integer (integral) sequences, because sequences are already sets. Best regard.

## Just wondering: Hofstadter sequences

I'm just wondering: has anyone here studied the Hofstadter sequences? (Such as the Q sequence: 1, 1, 2, 3, 3, 4, 5, 5, 6, 6, 6, 8, 8, 8, 10, 9, 10, 11, 11, 12, 12, 12, 12, 16, 14, 14, ...