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 "1, one 1, two 1s, one
2 one 1," etc., and the result is 1, 11, 21, 1211, 111221, .... Similarly, starting
the sequence instead with the digit  for  gives
 , 1 , 111 , 311 , 13211 , 111312211 , 31131122211 , 1321132132211 , ..., as summarized in the following table.
 | Sloane | sequence | | 1 | A005150 | 1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, ... | | 2 | A006751 | 2, 12, 1112, 3112, 132112, 1113122112,
311311222112, ... | | 3 | A006715 | 3, 13, 1113, 3113, 132113, 1113122113, 311311222113, ... |
The number of digits in the  th term of the sequence
for  are 1, 2, 2, 4, 6, 6, 8, 10, 14, 20,
26, 34, 46, 62, ... (Sloane's A005341). Similarly, the numbers of digits for the  th term of the sequence
for  , 3, ..., are 1, 2, 4, 4, 6, 10, 12,
14, 22, 26, ... (Sloane's A022471). These sequences are asymptotic to  , where
The quantity  is known as Conway's constant (Sloane's A014715), and amazingly is given by the unique positive real
root of the polynomial
 |
(4)
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all of whose roots are illustrated above.
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." It is postulated that every string of 1s, 2s, and 3s that
does not contain four of the same number in succession eventually "decays"
into a compound of 92 special elements, named after the chemical elements.
Conway, J. H. "The Weird and Wonderful Chemistry of Audioactive Decay."
Eureka 46, 5-18, 1986.
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.
Hilgemeier, M. "Die Gleichniszahlen-Reihe." Bild der Wissensch. 12,
194-196, Dec. 1986.
Hilgemeier, M. "'One Metaphor Fits All': A Fractal Voyage with Conway's Audioactive Decay." Ch. 7 in Pickover, C. A. (Ed.). Fractal Horizons: The Future Use of Fractals. New York:
St. Martin's Press, 1996.
Pickover, C. A. "Audioactive Decay." Ch. 55 in Wonders of Numbers: Adventures in Mathematics, Mind, and Meaning.
Oxford, England: Oxford University Press, pp. 134-138, 2001.
Sloane, N. J. A. Sequences A005150/M4780, A005341/M0321, A006715/M2965, and A006751/M2052 in "The On-Line Encyclopedia of Integer
Sequences."
Vardi, I. Computational Recreations in Mathematica. Reading,
MA: Addison-Wesley, pp. 13-14, 1991.
Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, p. 905,
2002.
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