## 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 A006751 and A006715. The number of Digits in the th term of both the sequences for is asymptotic to , where is a constant and

(Sloane's A014715) is Conway's Constant. is given by the largest Root of the Polynomial

In fact, the constant is even more general than this, applying to all starting sequences (i.e., even those starting with arbitrary starting digits), with the exception of 22, a result which follows from the Cosmological Theorem. Conway discovered that strings sometimes factor as a concatenation of two strings whose descendants never interfere with one another. A string with no nontrivial splittings is called an element,'' and other strings are called compounds.'' Every string of 1s, 2s, and 3s eventually decays'' into a compound of 92 special elements, named after the chemical elements.

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. Sequences A005150/M4780, A006715/M2965, and A006751/M2052 in 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.