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Conway's Constant

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Conway's constant

lambda=1.303577269034296...

(OEIS A014715) is the asymptotic growth factor of the number of digits in the look and say sequence: the number of digits in its nth term grows asymptotically as Clambda^n. It is the unique positive real 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)-x^(60)-x^(59)+2x^(58)+5x^(57)+3x^(56)-2x^(55)-10x^(54)-3x^(53)-2x^(52)+6x^(51)+6x^(50)+x^(49)+9x^(48)-3x^(47)-7x^(46)-8x^(45)-8x^(44)+10x^(43)+6x^(42)+8x^(41)-5x^(40)-12x^(39)+7x^(38)-7x^(37)+7x^(36)+x^(35)-3x^(34)+10x^(33)+x^(32)-6x^(31)-2x^(30)-10x^(29)-3x^(28)+2x^(27)+9x^(26)-3x^(25)+14x^(24)-8x^(23)-7x^(21)+9x^(20)+3x^(19)-4x^(18)-10x^(17)-7x^(16)+12x^(15)+7x^(14)+2x^(13)-12x^(12)-4x^(11)-2x^(10)+5x^9+x^7-7x^6+7x^5-4x^4+12x^3-6x^2+3x-6,

illustrated in the figure above. Note that 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, ... (OEIS A014967).

See also

Conway Sequence, Cosmological Theorem, Look and Say Sequence

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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. R. "Conway's Constant." §6.12 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 452-455, 2003.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 Fractal Horizons: The Future Use of Fractals (Ed. C. A. Pickover). New York: St. Martin's Press, 1996.Sloane, N. J. A. Sequences A014715 and A014967 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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Conway's Constant

Cite this as:

Weisstein, Eric W. "Conway's Constant." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ConwaysConstant.html

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