Aronson's sequence: Difference between revisions

Revision as of 08:48, 23 May 2012

Latest revision as of 13:52, 21 November 2022

(42 intermediate revisions by 17 users not shown)

Line 1:

{{Short description|Sequence of numbers}}

The '''Aronson's Sequence''' is a sequence of numbers that is defined as '''T is the first, fourth, eleventh, ... letter in this sentence, not counting spaces or commas'''.

'''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, 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 [[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=https://books.google.com/books?id=o8jzWF7rD6oC&pg=PA44}}.</ref>

==Origin==

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 sequence is based on the observation that ordinal numbers in the English language always end in "th".<ref>[http://everything2.com/title/Aronson%2527s+Sequence everything2.com]</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

The first few terms of the sequence are: 1,4,11,16,24,29,33......

| last1 = Cloitre | first1 = Benoit

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

==Recognition==

| last3 = Vandermast | first3 = Matthew J.

| arxiv = math/0305308

The sequence was mentioned in Douglas Hofstadter's book [[Metamagical Themas]].<ref>[http://mathworld.wolfram.com/AronsonsSequence.html wolfram.com]</ref>

| journal = Journal of Integer Sequences

| at = Art. 03.2.2

The sequence was assigned the number A005224 by the [[OEIS]].<ref>[http://oeis.org/A005224 oeis.org]</ref>

| title = Numerical analogues of Aronson's sequence

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

| volume = 6

| issue = 2003

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

== References ==

Line 17 ⟶ 24:

== External links ==

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

* http://oeis.org/A005224

{{Classes of natural numbers}}

* http://livearchive.onlinejudge.org/index.php?option=com_onlinejudge&Itemid=8&page=show_problem&problem=2891

[[Category:IntegerBase-dependent integer sequences]]

[[Category:Integer sequences]]