The lines show the growth of the numbers of digits in the look-and-say sequences with starting points 23 (red), 1 (blue), 13 (violet), 312 (green). These lines (when represented in a logarithmic vertical scale) tend to straight lines whose slopes coincide with Conway's constant.
1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, 31131211131221, ... (sequence A005150 in the OEIS).
To generate a member of the sequence from the previous member, read off the digits of the previous member, counting the number of digits in groups of the same digit. For example:
1 is read off as "one 1" or 11.
11 is read off as "two 1s" or 21.
21 is read off as "one 2, one 1" or 1211.
1211 is read off as "one 1, one 2, two 1s" or 111221.
111221 is read off as "three 1s, two 2s, one 1" or 312211.
The look-and-say sequence was analyzed by John Conway
after he was introduced to it by one of his students at a party.
The sequence grows indefinitely. In fact, any variant defined by starting with a different integer seed number will (eventually) also grow indefinitely, except for the degenerate sequence: 22, 22, 22, 22, … (sequence A010861 in the OEIS)
Digits presence limitationEdit
No digits other than 1, 2, and 3 appear in the sequence, unless the seed number contains such a digit or a run of more than three of the same digit.
Conway's cosmological theorem asserts that every sequence eventually splits ("decays") into a sequence of "atomic elements", which are finite subsequences that never again interact with their neighbors. There are 92 elements containing the digits 1, 2, and 3 only (92 is exactly the number of Johnson solids and exactly the number of non-transuranium elements), which John Conway named after the chemical elements up to uranium, calling the sequence audioactive. There are also two "transuranic" elements for each digit other than 1, 2, and 3.
Growth in lengthEdit
The terms eventually grow in length by about 30% per generation. In particular, if Ln denotes the number of digits of the n-th member of the sequence, then the limit of the ratio exists and is given by
where λ = 1.303577269034... (sequence A014715 in the OEIS) is an algebraic number of degree 71. This fact was proven by Conway, and the constant λ is known as Conway's constant. The same result also holds for every variant of the sequence starting with any seed other than 22.
This polynomial was correctly given in Conway's original Eureka article,
but in the reprinted version in the book edited by Cover and Gopinath the term was incorrectly printed with a minus sign in front.
The look-and-say sequence is also popularly known as the Morris Number Sequence, after cryptographer Robert Morris, and the puzzle "What is the next number in the sequence 1, 11, 21, 1211, 111221?" is sometimes referred to as the Cuckoo's Egg, from a description of Morris in Clifford Stoll's book The Cuckoo's Egg.
There are many possible variations on the rule used to generate the look-and-say sequence. For example, to form the "pea pattern" one reads the previous term and counts all instances of each digit, listed in order of their first appearance, not just those occurring in a consecutive block. So beginning with the seed 1, the pea pattern proceeds 1, 11 ("one 1"), 21 ("two 1s"), 1211 ("one 2 and one 1"), 3112 ("three 1s and one 2"), 132112 ("one 3, two 1s and one 2"), 311322 ("three 1s, one 3 and two 2s"), etc. This version of the pea pattern eventually forms a cycle with the two "atomic" terms 23322114 and 32232114.
Other versions of the pea pattern are also possible; for example, instead of reading the digits as they first appear, one could read them in ascending order instead. In this case, the term following 21 would be 1112 ("one 1, one 2") and the term following 3112 would be 211213 ("two 1s, one 2 and one 3").
These sequences differ in several notable ways from the look-and-say sequence. Notably, unlike the Conway sequences, a given term of the pea pattern does not uniquely define the preceding term. Moreover, for any seed the pea pattern produces terms of bounded length: This bound will not typically exceed 2 × Radix + 2 digits (22 digits for decimal: radix = 10) and may only exceed 3 × Radix digits (30 digits for decimal radix) in length for long, degenerate, initial seeds (sequence of "100 ones", etc.). For these extreme cases, individual elements of decimal sequences immediately settle into a permutation of the form a0 b1 c2 d3 e4 f5 g6 h7 i8 j9 where here the letters a–j are placeholders for digit counts from the preceding sequence element.
Since the sequence is infinite, and the length of each element is bounded, it must eventually repeat, due to the pigeonhole principle. As a consequence, pea pattern sequences are always eventually periodic.
^ abcConway, J. H. (January 1986). "The Weird and Wonderful Chemistry of Audioactive Decay" (PDF). Eureka. 46: 5–16.
Conway, J. H. (1987). "The Weird and Wonderful Chemistry of Audioactive Decay". In Cover, Thomas M.; Gopinath, B. (eds.). Open Problems in Communication and Computation. Springer-Verlag. pp. 173–188. ISBN 0-387-96621-8.
^Look-and-Say Numbers (feat John Conway) - Numberphile on YouTube
^Conway Sequence, MathWorld, accessed on line February 4, 2011.
Martin, Oscar (2006). "Look-and-Say Biochemistry: Exponential RNA and Multistranded DNA" (PDF). American Mathematical Monthly. Mathematical association of America. 113 (4): 289–307. doi:10.2307/27641915. ISSN 0002-9890. JSTOR 27641915. Archived from the original (PDF) on 2006-12-24. Retrieved January 6, 2010.
^Ekhad, S. B., Zeilberger, D.: Proof of Conway's lost cosmological theorem, Electronic Research Announcements of the American Mathematical Society, August 21, 1997, Vol. 5, pp. 78–82. Retrieved July 4, 2011.
Vardi, Ilan (1991). Computational Recreations in Mathematica. Addison-Wesley. ISBN 0-201-52989-0.