Aronson's sequence: Difference between revisions

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{{Short description|Sequence of numbers}}

'''Aronson's sequence''' is an [[integer sequence]] defined by the English sentence "T is the first, fourth, eleventh, sixteenth, ... letter in this sentence." Spaces and punctuation are ignored. The first few numbers in the sequence are: 1+1 2−2 3×3 4÷4 5⁵ ⁶6 7↑7

: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. The sequence is infinite—and this statement requires some proof. The proof depends on the observation that the English names of all [[ordinal number (linguistics)|ordinal number]]s, except those that end in 2, must contain at least one "t".<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/><ref>[http://arxiv.org/abs/math/0305308 "Numerical Analogues of Aronson's Sequence" by Benoit Cloitre, N. J. A. Sloane, Matthew J. Vandermast, Cornell University Library, 21 May 21, 2003]</ref>

{{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

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

| last1 = Cloitre | first1 = Benoit

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

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| 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]]