Conway's Constant

The constant

$$\lambda=1.303577269034296\ldots$$

(Sloane's A014715) giving the asymptotic rate of growth $C\lambda^k$ of the number of Digits in the $k$th term of the Look and Say Sequence. $\lambda$ is given by the largest Root of the Polynomial

$0 = x^{71} -x^{69}-2x^{68}-x^{67}+2x^{66}+2x^{65}+x^{64}-x^{63}-x^{62}-x^{61}$
$\quad -x^{60}-x^{59}+2x^{58}+5x^{57}+3x^{56}-2x^{55}-10x^{54}$
$\quad -3x^{53}-2x^{52}+6x^{51}+6x^{50}+x^{49}+9x^{48}-3x^{47}$
$\quad -7x^{46}-8x^{45}-8x^{44}+10x^{43}+6x^{42}+8x^{41}-4x^{40}$
$\quad -12x^{39}+7x^{38}-7x^{37}+7x^{36}+x^{35}-3x^{34}+10x^{33}$
$\quad +x^{32}-6x^{31}-2x^{30}-10x^{29}-3x^{28}+2x^{27}+9x^{26}$
$\quad -3x^{25}+14x^{24}-8x^{23}-7x^{21}+9x^{20}-3x^{19}-4x^{18}$
$\quad -10x^{17}-7x^{16}+12x^{15}+7x^{14}+2x^{13}-12x^{12}$
$\quad -4x^{11}-2x^{10}-5x^9+x^7-7x^6+7x^5-4x^4+12x^3-6x^2+3x-6.$	(1)

The Polynomial given in Conway (1987, p. 188) contains a misprint. The Continued Fraction for $\lambda$ is 1, 3, 3, 2, 2, 54, 5, 2, 1, 16, 1, 30, 1, 1, 1, 2, 2, 1, 14, 1, ... (Sloane's A014967).

See also Conway Sequence, Cosmological Theorem, Look and Say Sequence

References

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.

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/cnwy/cnwy.html

Sloane, N. J. A. A014715 and A014967 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.

Vardi, I. Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, pp. 13-14, 1991.