Editing Look-and-say sequence (section)

{{Short description|Integer sequence}}
{{Redirect|Look-and-say|the method for learning to read|look-and-say method}}
[[File:Conway's constant.svg|thumb|300px|The lines show the growth of the numbers of digits in the look-and-say sequences with starting points 23 (red), 1 (blue), 13 (violet), 312 (green). These lines (when represented in a [[logarithmic scale|logarithmic vertical scale]]) tend to straight lines whose slopes coincide with Conway's constant.]]

In [[mathematics]], the '''look-and-say sequence''' is the [[integer sequence|sequence of integers]] beginning as follows:

: 1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, 31131211131221, ... {{OEIS|id=A005150}}.

To generate a member of the sequence from the previous member, read off the digits of the previous member, counting the number of digits in groups of the same digit.  For example:

* 1 is read off as "one 1" or 11.
* 11 is read off as "two 1s" or 21.
* 21 is read off as "one 2, one 1" or 1211.
* 1211 is read off as "one 1, one 2, two 1s" or 111221.
* 111221 is read off as "three 1s, two 2s, one 1" or 312211.

The look-and-say sequence was analyzed by [[John Horton Conway|John Conway]]<ref name="Conway-original-article">
{{cite journal
|last=Conway
|first=John H.
|author-link=John Horton Conway
|title=The Weird and Wonderful Chemistry of Audioactive Decay
|journal=Eureka
|date=January 1986
|volume=46
|pages=5–16
|url=https://sites.math.rutgers.edu/~zeilberg/EM12/ConwayWW.pdf}}
Reprinted as
{{cite book
 |last=Conway
 |first=J. H.
 |author-link=John Horton Conway
 |editor-last=Cover
 |editor-first=Thomas M.
 |editor-last2=Gopinath
 |editor-first2=B.
 |title=Open Problems in Communication and Computation
 |publisher=[[Springer-Verlag]]
 |date=1987
 |pages=173–188
 |chapter=The Weird and Wonderful Chemistry of Audioactive Decay
 |isbn=0-387-96621-8}}
</ref>
after he was introduced to it by one of his students at a party.<ref>
{{Cite book
  | last = Roberts
  | first = Siobhan
  | authorlink = Siobhan Roberts
  | title = Genius at Play: The Curious Mind of John Horton Conway
  | publisher = [[Bloomsbury Publishing|Bloomsbury]]
  | year = 2015
  | isbn = 978-1-62040-593-2
}}
</ref><ref>
{{YouTube|id=ea7lJkEhytA|title=Look-and-Say Numbers (feat John Conway) - Numberphile}}
</ref>

The idea of the look-and-say sequence is similar to that of [[run-length encoding]].

If started with any digit ''d'' from 0 to 9 then ''d'' will remain indefinitely as the last digit of the sequence. For any ''d'' other than 1, the sequence starts as follows:
: ''d'', 1''d'', 111''d'', 311''d'', 13211''d'', 111312211''d'', 31131122211''d'', …
Ilan Vardi has called this sequence, starting with ''d'' = 3, the '''Conway sequence''' {{OEIS|id=A006715}}. (for ''d'' = 2, see {{oeis|id=A006751}})<ref>[http://mathworld.wolfram.com/ConwaySequence.html Conway Sequence], [[MathWorld]], accessed on line February 4, 2011.</ref>