This is known as
the Look and Say Sequence and was pioneered by and related to much mathematics done by
Conway.
nieken's got it right in describing it, and the next few terms would be:
1
11
21
1211
111221
312211
13112221
1113213211
31131211131221
13211311123113112211
11131221133112132113212221
3113112221232112111312211312113211
. . .
If we had started the sequence with another digit 'd' (instead of 1) where 2 =< d =< 9, then the sequence would have read:
d
1d
111d
311d
13211d
111312211d
. . .
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:
1
2
2
4
6
6
8
10
14
. . .
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