So, no one I have asked has been able to find the pattern here. Can you?

Question

1, 11, 21, 1211, 111221...

Can you find the next number in the sequence?

Answer 1

This is called the "Look-and-Say" sequence. It was invented by the mathematician John Horton Conway.

The first number is 1.

To get the second number, say what you see--"One one"--and then write it: 11.

To get the third number, say what you see--"Two ones"--and then write it: 21.

To get the fourth number, say what you see--"One two, one one"--and then write it: 1211.

To get the fifth number, say what you see--"One one, one two, two ones"--and then write it: 111221.

Thus, the next number is 312211. And so on.

Answer 2

This is not really a mathematical sequence. It is a look and say sequence.

The next term is 312211.
That is to say, when you look at the previous number 111221 as a sequence of digits you have:
three ones 31
followed by two twos 22
followed by one one 11
Put it together and you get 312211.

Answer 3

A friend of mine put this on his final. Read each line out loud. Each line refers to the line above.
1: one
11: one 1
21: two 1's
1211: one 2, one 1
111221: one 1, one 2, two 1's
Next line: 312211
then: 13112221,
etc

Answer 4

312211

Answer 5

Yeah, what they said!