Look-and-say Sequence - Basic Properties

Basic Properties

• 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, ….
• 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: Every sequence eventually splits 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 natural chemical elements. There are also two "transuranic" elements for each digit other than 1,2, and 3.
• The terms eventually grow in length by about 30% per generation. In particular, if L_n 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 = 1.303577269... 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.

Conway's constant is the unique positive real root of the following polynomial: