thumb|300px|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 scale|logarithmic vertical scale) tend to straight lines whose slopes coincide with Conway's constant.]]
In mathematics, the look-and-say sequence is the sequence of integers beginning as follows:
: 1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, 31131211131221, ... .
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 idea of the look-and-say sequence is similar to that of run-length encoding.
If started with any digit d from 0 to 9 then d will remain indefinitely as the last digit of the sequence. For any d other than 1, the sequence starts as follows:
: d, 1d, 111d, 311d, 13211d, 111312211d, 31131122211d, …
Ilan Vardi has called this sequence, starting with d = 3, the Conway sequence . (for d = 2, see )
Basic properties
frame|Roots of the Conway polynomial plotted in the [[complex plane. Conway's constant is marked with the Greek letter lambda (λ).]]
Growth
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, ... which remains the same size.
Cosmological decay
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, which John Conway named after the 92 naturally occurring chemical elements up to uranium, calling the sequence "audioactive." There are also two "transuranic" elements (Np and Pu) for each digit other than 1, 2, and 3. Below is a table of all such elements:
{| class="wikitable mw-collapsible mw-collapsed"
! colspan="5" |All "atomic elements" (Where E<sub>k</sub> is included within the derivate of E<sub>k+1</sub> except Np and Pu) This fact was proven by Conway, and the constant λ is known as Conway's constant.
Popularization
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.
Variations
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. Since the sequence is infinite, the length of each element in the sequence is bounded, and there are only finitely many words that are at most a predetermined length, it must eventually repeat, and as a consequence, pea pattern sequences are always eventually periodic.