PlanetMath: sequence

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sequence

(Definition)

Sequences

Given any set , a sequence in is a function $f\colon \mathbb{N} \to X$ from the set of natural numbers to . 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 , a generalized sequence or transfinite sequence in 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.

"sequence" is owned by djao.

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See Also: convergent sequence

Also defines:

generalized sequence, transfinite sequence, finite sequence

Attachments:: geometric sequence (Definition) by pahio
power tower sequence (Example) by pahio

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Cross-references: finite, ordinal, subscript, natural numbers, function
There are 645 references to this entry.

This is version 6 of sequence, born on 2001-10-19, modified 2008-04-22.
Object id is 397, canonical name is Sequence.
Accessed 27553 times total.

Classification:

AMS MSC:	40-00 (Sequences, series, summability :: General reference works )
	03E10 (Mathematical logic and foundations :: Set theory :: Ordinal and cardinal numbers)

Pending Errata and Addenda

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Discussion

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Just wondering: Hofstadter sequences by PrimeFan on 2007-12-05 20:12:41

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, ...

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What is sequence anyway by XJamRastafire on 2002-06-19 19:56:32

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.

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