Just say it!

If you interpret the elements of the sequence as numerical quantities, there seems to be no obvious pattern. But who said that they are numbers? If you look at the relationship between an element and its predecessor and focus on “symbolic” content, we see a pattern. Each element “describes” the previous one. For example, the third element is $21$ , which can be described as “one 2 and one 1,” i.e., $1211$ , which is the fourth element. This can be described as “one 1, one 2, and two 1s,”i.e., $111221$ . So the next member is $312211$ (“three 1s, two 2s, and one 1”).

$\large\text{Speak about your friend on your left!}$

$1, x$

It has one "one"

$1, 11, x$

It has two "one" s

$1, 11, 21, x$

It has one "two" and one "one"

$1, 11, 21, 1211, x$

It has one "one" , one "two" , and two "one" s again

$1, 11, 21, 1211, 111221, x$

It has three "one" s two "two" es, and one "one" again

$1, 11, 21, 1211, 111221, \boxed{312211}$

$\text{Therefore, the next number in this sequence should be }\boxed{312211}$