Look and Say Sequence
The Integer Sequence beginning with a single digit in which the next term is obtained by describing the previous term.Starting with 1, the sequence would be defined by ``one 1, two 1s, one 2 two 1s,'' etc., and the result is 1, 11, 21,1211, 111221, 312211, 13112221, 1113213211, ... (Sloane's A005150).
Starting the sequence instead with the digit 
for 
gives , 1, 111, 311, 13211, 111312211,31131122211, 1321132132211, ... The sequences for 
and 3 are Sloane's A006751and A006715. The number ofDigits in the 
th term of both the sequences for 
is asymptotic to 
, where 
is aconstant and
(Sloane's A014715) is Conway's Constant.
is given by the largest Root of the Polynomial | |
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In fact, the constant is even more general than this, applying to all starting sequences (i.e., even thosestarting with arbitrary starting digits), with the exception of 22, a result which follows from the CosmologicalTheorem. Conway discovered that strings sometimes factor as a concatenation of two strings whose descendants neverinterfere with one another. A string with no nontrivial splittings is called an ``element,'' and other strings arecalled ``compounds.'' Every string of 1s, 2s, and 3s eventually ``decays'' into a compound of 92 special elements,named after the chemical elements.
See also Conway's Constant, Cosmological Theorem
References
Conway, J. H. ``The Weird and Wonderful Chemistry of Audioactive Decay.'' Eureka 45, 5-18, 1985. Conway, J. H. ``The Weird and Wonderful Chemistry of Audioactive Decay.'' §5.11 in Open Problems in Communications and Computation. (Ed. T. M. Cover and B. Gopinath). New York: Springer-Verlag, pp. 173-188, 1987. Conway, J. H. and Guy, R. K. ``The Look and Say Sequence.'' In The Book of Numbers. New York: Springer-Verlag, pp. 208-209, 1996. Sloane, N. J. A. SequencesA005150/M4780,A006715/M2965, andA006751/M2052in ``An On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995. Vardi, I. Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, pp. 13-14, 1991.
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