Aronson's sequence: Difference between revisions

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{{Short description|Sequence of numbers}}

'''Aronson's sequence''' is aan [[integer sequence of numbers that is]] defined to makeby the English sentence "T is the first, fourth, eleventh, sixteenth, ... letter in this sentence,." not countingSpaces spacesand orpunctuation commas"are trueignored. The first few numbers in the sequence are:

:1, 4, 11, 16, 24, 29, 33, 35, 39, 45, 47, 51, 56, 58, 62, 64, 69, 73, 78, 80, 84, 89, 94, 99, 104, 111, 116, 122, 126, 131, 136, 142, 147, 158, 164, 169, ... {{OEIS|A005224}}.

In [[Douglas Hofstadter]]'s book ''[[Metamagical Themas]]'', the sequence is credited to J. K.[[Jeffrey Aronson]] of Oxford, England;. itThe sequence is basedinfinite—and this statement requires some proof. The proof depends on the observation that the English names of all [[Ordinalordinal number (linguistics)|ordinal numbersnumber]]s, except those that end in the2, Englishmust languagecontain alwaysat endleast inone "tht".<ref>{{citation|title=Metamagical Themas: Questing For The Essence Of Mind And Pattern|first=Douglas R.|last=Hofstadter|authorlink=Douglas Hofstadter|publisher=Basic Books|year=1996|isbn=9780465045662|page=44|url=httphttps://books.google.com/books?id=o8jzWF7rD6oC&pg=PA44}}.</ref>

Aronson's sequence is closely related to [[autogram]]s. There are many generalizations of Aronson's sequence and research into the topic is ongoing.<ref name=benoit/>

The Aronson's sequence is essentially an [[Autogram]] that describes itself.

however{{harvtxt|Cloitre|Sloane|Vandermast|2003}} write that Aronson's sequence is "a classic example of a [[Self-reference|self-referential]] sequence." However, they criticize it for being ambiguously defined due to the variation in naming of numbers over one hundred in different dialects of English. In its place, they offer several other self-referential sequences whose definitions rely only on mathematics rather than on the English language.<ref name=benoit>{{citation

The first few numbers in the sequence are:

:1, 4, 11, 16, 24, 29, 33, ... {{OEIS|A005224}}.

{{harvtxt|Cloitre|Sloane|Vandermast|2003}} write that Aronson's sequence is "a classic example of a [[Self-reference|self-referential]] sequence";

however, they criticize it for being ambiguously defined due to the variation in naming of numbers over one hundred in different dialects of English. In its place, they offer several other self-referential sequences whose definitions rely only on mathematics rather than on the English language.<ref>{{citation

| last1 = Cloitre | first1 = Benoit

| last2 = Sloane | first2 = N. J. A. | author2-link = Neil Sloane

Line 17 ⟶ 15:

| at = Art. 03.2.2

| title = Numerical analogues of Aronson's sequence

| url = http://www.emis.ams.orgde/journals/JIS/VOL6/Cloitre/cloitre2.pdf

| volume = 6

| yearissue = 2003}}.</ref>

| year = 2003| bibcode = 2003JIntS...6...22C}}.</ref>

== References ==

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== External links ==

* {{mathworld|urlname=AronsonsSequence|title=Aronson's Sequence}}

{{Classes of natural numbers}}

[[Category:Base-dependent integer sequences]]