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 $d$ for $2\leq d\leq 9$ gives $d$, 1$d$, 111$d$, 311$d$, 13211$d$, 111312211$d$, 31131122211$d$, 1321132132211$d$, ... The sequences for $d=2$ and 3 are Sloane's A006751 and A006715. The number of Digits in the $n$th term of both the sequences for $1\leq n\leq 9$ is asymptotic to $C\lambda^n$, where $C$ is a constant and

$$\lambda=1.303577269034296\ldots$$

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

$0 = x^{71} -x^{69}-2x^{68}-x^{67}+2x^{66}+2x^{65}+x^{64}-x^{63}-x^{62}-x^{61}$
$\quad -x^{60}-x^{59}+2x^{58}+5x^{57}+3x^{56}-2x^{55}-10x^{54}$
$\quad -3x^{53}-2x^{52}+6x^{51}+6x^{50}+x^{49}+9x^{48}-3x^{47}$
$\quad -7x^{46}-8x^{45}-8x^{44}+10x^{43}+6x^{42}+8x^{41}-4x^{40}$
$\quad -12x^{39}+7x^{38}-7x^{37}+7x^{36}+x^{35}-3x^{34}+10x^{33}$
$\quad +x^{32}-6x^{31}-2x^{30}-10x^{29}-3x^{28}+2x^{27}+9x^{26}$
$\quad -3x^{25}+14x^{24}-8x^{23}-7x^{21}+9x^{20}-3x^{19}-4x^{18}$
$\quad -10x^{17}-7x^{16}+12x^{15}+7x^{14}+2x^{13}-12x^{12}$
$\quad -4x^{11}-2x^{10}-5x^9+x^7-7x^6+7x^5-4x^4+12x^3-6x^2+3x-6.$

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.

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. 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.