*

The essence sequence start with a solitary digit in i beg your pardon the following term is acquired by relenten the previous term. Starting with 1, the sequence would be defined by "1, one 1, 2 1s, one 2 one 1," etc., and the result is 1, 11, 21, 1211, 111221, .... Similarly, starting the sequence rather with the digit

*
for
*
1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, ...

You are watching: 1,11,21,1211,111221,312211

2A0067512, 12, 1112, 3112, 132112, 1113122112, 311311222112, ...3A0067153, 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, ... (OEIS A005341). Similarly, the numbers of digits for the
*
th hatchet of the sequence because that
*
, 3, ..., are 1, 2, 4, 4, 6, 10, 12, 14, 22, 26, ... (OEIS A022471). This sequences space asymptotic come
*
, where

*
*
*

(1)
*
*
*

(2)
*
*
*

(3)
*

The quantity

*
is well-known as Conway"s constant (OEIS A014715), and also amazingly is given by the unique positive actual root the the polynomial


*

(4)

all of who roots are depicted above.

In fact, the constant is even much more general than this, using to all beginning sequences (i.e., even those starting with arbitrary beginning digits), through the exemption of 22, a result which adheres to from the cosmological theorem. Conway uncovered that strings sometimes factor as a concatenation of 2 strings whose descendants never ever interfere through one another. A string v no nontrivial splittings is called an "element," and also other strings are dubbed "compounds." it is postulated the every string of 1s, 2s, and also 3s the does not contain four of the exact same number in succession eventually "decays" right into a link of 92 distinct elements, named after the chemical elements.


SEE ALSO: Conway"s Constant, Cosmological Theorem, Run-Length Encoding
REFERENCES:

Conway, J.H. "The Weird and also Wonderful Chemistry the 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 also Computation. (Ed. T.M.Cover and also B.Gopinath). New York: Springer-Verlag, pp.173-188, 1987.

Conway, J.H. And Guy, R.K. "The Look and also Say Sequence." In The book of Numbers. Brand-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 through Conway"s Audioactive Decay." Ch.7 in Pickover, C.A. (Ed.). Fractal Horizons: The Future usage of Fractals. Brand-new York: St.Martin"s Press, 1996.

Pickover, C.A. "Audioactive Decay." Ch.55 in marvels of Numbers: Adventures in Mathematics, Mind, and Meaning. Oxford, England: Oxford college Press, pp.134-138, 2001.

Sloane, N.J.A. Assignment A005150/M4780, A005341/M0321, A006715/M2965, and also A006751/M2052 in "The On-Line Encyclopedia of integer Sequences."

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

ptcouncil.net, S. A new Kind that Science. Champaign, IL: ptcouncil.net Media, p.905, 2002.


Referenced top top ptcouncil.net|Alpha: Look and also Say Sequence
CITE THIS AS:

Weisstein, Eric W. "Look and also Say Sequence."From ptcouncil.net--A ptcouncil.net web Resource. Https://ptcouncil.net/LookandSaySequence.html


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