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.

© 1996-9

1999-05-25