medium | general |

What is/are the next term(s) in the sequence: a) 1, 11, 21, 1211, 111221, ? b) 10, 11, 12, 13, 14, 20, 22, 101, ?, ? c) (This a sequence made by only 2 & 1): 2,2,1,1,2,1,2,2,1, ?, ?....

a) base-change, b) count numbers, c) count repetition

a) 1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, … This is a Look-and-say sequence! To generate a member of the sequence from the previous member, read off the digits of the previous member, counting the number of digits in groups of the same digit.

b) 10, 11, 12, 13, 14, 20, 22, 101, 1010, 1111111111 This is '10' written in different bases, from 10 to 1!

c) This a sequence made by only 2 & 1: 2,2,1,1,2,1,2,2,1, ?, ?.... This is a version of Kolakoski sequence, and is its own run-length encoding. Each symbol occurs in a "run" of either 1 or 2 consecutive terms, and writing down the lengths of these runs gives exactly the same sequence. It is the unique sequence with this property except for the same sequence with extra '1' at start.