def las(s):
    out=[]; i=0; n=len(s)
    while i<n:
        j=i
        while j<n and s[j]==s[i]: j+=1
        out.append(str(j-i)); out.append(s[i]); i=j
    return ''.join(out)

def run(seed, steps, cap=8_000_000):
    s=seed; lens=[len(s)]
    for _ in range(steps):
        s=las(s)
        lens.append(len(s))
        if len(s)>cap: break
    return s, lens

CONWAY=1.303577269034296
for seed in ["1","3","2","1112"]:
    s,lens=run(seed,55)
    ratios=[lens[i+1]/lens[i] for i in range(len(lens)-1)]
    tail=ratios[-3:]
    print("seed=%-5s steps=%2d  final ratio=%.6f  |lambda-r|=%.2e  digits=%s"%(
        seed, len(lens)-1, ratios[-1], abs(ratios[-1]-CONWAY), ''.join(sorted(set(s)))))
# the '22' exception: it is a fixed point (stays length 2 forever)
s,lens=run("22",10)
print("seed=22  (the known exception) lengths:", lens)