Aronson's sequence is an
integer sequence
In mathematics, an integer sequence is a sequence (i.e., an ordered list) of integers.
An integer sequence may be specified ''explicitly'' by giving a formula for its ''n''th term, or ''implicitly'' by giving a relationship between its terms. For ...
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, ... .
In
Douglas Hofstadter
Douglas Richard Hofstadter (born 15 February 1945) is an American cognitive and computer scientist whose research includes concepts such as the sense of self in relation to the external world, consciousness, analogy-making, Strange loop, strange ...
's book ''
Metamagical Themas'', the sequence is credited to
Jeffrey Aronson
Jeffrey Kenneth Aronson (born 1947) is a British physician and clinical pharmacologist, currently working at the Centre for Evidence-Based Medicine in Oxford.
Education
Aronson studied at the University of Glasgow from 1964 to 1970, qua ...
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
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets.
A finite set can be enumerated by successively labeling each element with the leas ...
s, except those that end in 2, must contain at least one "t".
[.]
Aronson's sequence is closely related to
autogram
An autogram ( = self, = letter) is a sentence that describes itself in the sense of providing an inventory of its own characters. They were invented by Lee Sallows, who also coined the word ''autogram''. An essential feature is the use of full ca ...
s. There are many generalizations of Aronson's sequence and research into the topic is ongoing.
[
write that Aronson's sequence is "a classic example of a ]self-referential
Self-reference is a concept that involves referring to oneself or one's own attributes, characteristics, or actions. It can occur in language, logic, mathematics, philosophy, and other fields.
In natural language, natural or formal languages, ...
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.[.]
References
External links
*
{{Classes of natural numbers
Base-dependent integer sequences