The look-and-say sequence starts with 1. Each term describes the digit runs in the previous term: 1 becomes '1 one' = 11. Then '2 ones' = 21. Then '1 two, 1 one' = 1211. Then '1 one, 1 two, 2 ones' = 111221. Then 312211. The lengths of these terms grow, and their ratio converges to λ ≈ 1.30357.
John Conway analysed look-and-say sequences in his 1986 paper and proved the Cosmological Theorem: every look-and-say sequence, after at most 24 steps, breaks into a fixed collection of 92 subsequences he called 'atoms' or 'elements'. These 92 elements behave like chemical elements, each evolving independently. The growth rate of every such sequence is the same constant λ.
λ is the largest real root of a specific degree-71 polynomial with integer coefficients, derived by Conway from the characteristic polynomial of the matrix describing how the 92 atoms interact. It is algebraic by construction. The sequence never contains a digit higher than 3, a fact Conway also proved.
Lengths grow irregularly at first but the long-run ratio settles at λ ≈ 1.304. After the sequence splits into its 92 atomic subsequences (within 24 steps), every atom grows at exactly this rate.
After at most 24 steps, any look-and-say sequence decomposes into a combination of 92 subsequences Conway named after chemical elements (H, He, Li...). Each atom evolves independently, and each grows at rate λ. The total length is the sum of all atom lengths, growing at λ.
Lambda is the largest real root of a specific degree-71 polynomial with integer coefficients, derived from the recurrence relations between Conway's 92 atomic subsequences. It is therefore algebraic, not transcendental. This polynomial was computed by Conway and is one of the largest minimal polynomials for any naturally arising constant.
The Conway constant lambda ≈ 1.30357 is the growth rate of the look-and-say sequence 1, 11, 21, 1211, 111221, 312211... John Conway proved in 1986 that after at most 24 steps, any look-and-say sequence breaks into 92 fixed atomic subsequences. Every such sequence grows at exactly rate lambda. Uniquely among naturally arising constants, lambda is algebraic: the largest real root of a specific degree-71 polynomial.