This is known as the Look and Say Sequence
and was pioneered by and related to much mathematics done by Conway
's got it right in describing it, and the next few terms would be:
. . .
If we had started the sequence with another digit 'd' (instead of 1) where 2 =< d =< 9, then the sequence would have read:
. . .
An interesting and related problem is that of how many digits are in then nth term in the sequence. After counting, we notice that it is:
. . .
This sequence is asymptotic to C*lamda^n where lamda is Conway's Constant (approx equal to 1.30357...) and is given by the unique positive real root of the (behemoth) equation:
0 = 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 - 4x^40 - 12x^39
+ 7x^38 - 7x^37 + 7x^36 + x^35 - 3x^34 + 10x^33 + x^32 - 6x^31 - 2x^30 - 0x^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