Each sequence is read aloud to produce the next: "1" → "one 1" → "11". Lengths grow irregularly at first but the ratio stabilises at λ ≈ 1.304 (Conway's constant).
The ratio of consecutive string lengths oscillates but converges to Conway's constant λ ≈ 1.30358.
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.