What is this sequence? 1, 11, 21, 22211, 33221

Question

In this sequence:

1, 11, 21, 22211, 33221, 433332211, 44444444332221, ?

Can you guess the next number? What is this sequence?

Hint 1:

Every number always ends with 1 or 11

Hint 2:

If the number ends with 11, Then the next number ends with 1111 but this is longer, So ends with 1

Answer 1

The next number is:

5555554444433333211

Here is the process to transform a number to find the next one in the sequence :

Consider each value ($1$'s,$2$'s,$3$'s,...) separately. Per construction, they will always be written in decreasing order. Then:
* First each value $k$ is repeated $k+1$ times: $1$ becomes $11$, $2$ becomes $222$, $3$ becomes $3333$, etc.
* Then if a value $k$ is appearing $k+2$ times or more, each block of $k+2$ occurences is replaced by one occurence of $k+1$

Hence:

$1$ becomes

$11$ and nothing is replaced

$11$ becomes

$1111$ and the first three $1$'s are replaced by one $2$, making $21$

$21$ becomes

$22211$ and nothing is replaced.

$22211$ becomes

$2222222221111$ ; Then the first eight $2$'s are replaced by $33$, the first three $1$'s are replaced by one $2$, making $33221$

$33221$ becomes

first $3333333322222211$ ; and then $433332211$ after replacements of five $3$'s by one $4$ and four $2$'s by one $3$.

$433332211$ becomes

first $444443333333333333333221111$ ; and then $44444444332221$ after replacements

$44444444332221$ becomes

first $44444444444444444444444444444444444444443333333322222222211$ ; and then $5555554444433333211$ after replacements.

Research process on paper:

my paper