Fast look-and-say computation (for the binary case, starting with 0), implemented in Sage.

look_and_say.sage

import itertools

M = {}
 
def look_and_say(atom,n) :
  global M
  a = atom
  if n == 0 :
    return 0
  b = ""
  for c,l in itertools.groupby(a) :
    b += bin(len(list(l)))[2:] + c
  if n == 1 :
    return len(b)
  P = (atom,n)
  if P in M :
    return M[P]
  a = b + "2"
  start = 0
  prev = False
  atom = ""
  ans = 0
  for j,s in enumerate(a) :
    if s == "1" and prev  :
      ans += look_and_say(a[start:j],n-1)
      start = j
      prev = False
    elif s == "0" :
      prev = True
    elif s == "2" :
      ans += look_and_say(a[start:j],n-1)
  M[P] = ans
  return ans

R = RealField(1000)
 
print R(look_and_say("0",700)) / R(look_and_say("0",699))