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:
float(realroots(x^71 -x^69-2*x^68-x^67+2*x^66+2*x^65+x^64-x^63-x^62-x^61-x^60 -x^59+2*x^58+5*x^57+3*x^56-2*x^55-10*x^54-3*x^53-2*x^52+6*x^51+6*x^50 +x^49+9*x^48-3*x^47-7*x^46-8*x^45-8*x^44+10*x^43+6*x^42+8*x^41-5*x^40 -12*x^39+7*x^38-7*x^37+7*x^36+x^35-3*x^34+10*x^33+x^32-6*x^31-2*x^30 -10*x^29-3*x^28+2*x^27+9*x^26-3*x^25+14*x^24-8*x^23-7*x^21+9*x^20 +3*x^19-4*x^18-10*x^17-7*x^16+12*x^15+7*x^14+2*x^13-12*x^12-4*x^11-2*x^10 +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