Basic properties Look-and-say sequence

1 basic properties

1.1 growth
1.2 digits presence limitation
1.3 cosmological decay
1.4 growth in length

1.4.1 polynomial returning conway s constant

basic properties



growth

the sequence grows indefinitely. in fact, variant defined starting different integer seed number (eventually) grow indefinitely, except degenerate sequence: 22, 22, 22, 22, … (sequence a010861 in oeis)

digits presence limitation

no digits other 1, 2, , 3 appear in sequence, unless seed number contains such digit or run of more 3 of same digit.

cosmological decay

conway s cosmological theorem asserts every sequence splits ( decays ) sequence of atomic elements , finite subsequences never again interact neighbors. there 92 elements containing digits 1, 2, , 3 only, john conway named after chemical elements uranium, calling sequence audioactive. there 2 transuranic elements each digit other 1, 2, , 3.

growth in length

the terms grow in length 30% per generation. in particular, if ln denotes number of digits of n-th member of sequence, limit of ratio






l

n
+
1



l

n






{\displaystyle {\frac {l_{n+1}}{l_{n}}}}

exists , given by

lim

n
→
∞





l

n
+
1



l

n




=
λ


{\displaystyle \lim _{n\to \infty }{\frac {l_{n+1}}{l_{n}}}=\lambda }

where λ = 1.303577269034... (sequence a014715 in oeis) algebraic number of degree 71. fact proven conway, , constant λ known conway s constant. same result holds every variant of sequence starting seed other 22.

polynomial returning conway s constant

conway s constant unique positive real root of following polynomial: (sequence a137275 in oeis)

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






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

in original article, conway gives incorrect value polynomial, writing - instead of + in front of




x

35




{\displaystyle x^{35}}

. however, value of λ given in article correct.