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Look-and-Say Sequence (#107)

Jonathan Frech,

An interesting nonmathematical sequence which yet has interesting math­e­mat­i­cal prop­er­ties is the Look-and-say sequence.
As every good sequence does, it starts with a 1. From there you get each next entry by looking at the previous one and saying it. The next entry will thus be “one 1”, or 11. From there, “two 1”, or 21, then “one 2; one 1”, or 1211, “one 1; one 2; two 1”, or 111221, and so on.

Gen­er­ating the Sequence

The math­e­mat­i­cal prop­er­ty, although, does not lie in the entry’s value, but rather in its length.

Cal­cu­lat­ing the Length of each Entry

The interesting thing about the length is that it ap­prox­i­mate­ly grows by 𝜆 each time. $\big(\lambda=1.303577269034\dots\big)$$\big(\lambda=1.303577269034\dots\big)$$\big(\lambda=1.303577269034\dots\big)$
This con­stant 𝜆 can be calculated by getting the real root of a degree 71 polynomial (see Nathaniel Johnston’s post for further in­for­ma­tion) or be approximated by dividing entries. $\big(\lambda=\frac{\text{entry}_{n+1}}{\text{entry}_{n}}\text{ for }n\rightarrow\infty\big)$$\big(\lambda=\frac{\text{entry}_{n+1}}{\text{entry}_{n}}\text{ for }n\rightarrow\infty\big)$$\big(\lambda=\frac{\text{entry}_{n+1}}{\text{entry}_{n}}\text{ for }n\rightarrow\infty\big)$

Approximating 𝜆

Source code: look-and-say-sequence.py