1, 11, 21, 1211, 111221, 312211, 13112221, …

And the next element of the sequence is…


Just read 13112221 as “one 1, one 3, two 1s, three 2s, one 1”. This is look-and-say sequence (A005150 in the OEIS).

Each next number is about 30% longer. More precisely, if denotes the number of digits of the -th member of the sequence, then

One can enumerate 92 possible subsequences, construct a transition matrix, interpret it as a recurrence relation for the length of terms in the look-and-say sequence, recall that limiting ratio of terms in the sequence is equal to the spectral radius of the transition matrix, and finally find , the only positive rational root of Conway’s polynomial


using a computer algebra system like wxMaxima:

+5*x^9+x^7-7*x^6+7*x^5-4*x^4+12*x^3-6*x^2+3*x-6, 1e-8));

Source: Nathaniel Johnston