Look and Say sequence
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In mathematics, the look and say sequence is the sequence of integers beginning as follows:
- 1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, ...
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" that is, 11.
- 11 is read off as "two 1's" that is, 21.
- 21 is read off as "one 2, then one 1" that is, 1211.
- 1211 is read off as "one 1, then one 2, then two 1's" that is, 111221.
- 111221 is read off as "three 1, then two 2, then one 1" that is 312211.
The idea is similar to that of run-length encoding.
Properties
This sequence was first introduced by John Conway in 1987 under the name "audioactive decay". In the same paper Conway also proved that, if <math>L_i<math> is the length of the sequence on the <math>i<math>th iteration:
- <math>\frac{L_{i+1}}{L_{i}} \rightarrow \lambda<math>
where <math>\lambda = 1.303577269\ldots<math> is an algebraic number of degree 71 known as Conway's constant. This property also holds for all variants of the Look and Say sequence defined by beginning with a different number (e.g. 13, 1113, 3113, 132113...), except for the degenerate case 22, 22, 22, 22....
External links
- On-Line Encyclopedia of Integer Sequences (http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A005150)
- MathWorld (http://mathworld.wolfram.com/LookandSaySequence.html)
- Look and Say sequence generator (http://www.btinternet.com/~se16/js/looknsay.htm)