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

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

• aw it reminds me of oeis.org/A005150 Dec 30, 2021 at 0:12
• @BenjaminWang this is not look and say sequence Dec 30, 2021 at 0:18

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:

• How do you found the solution? Dec 30, 2021 at 13:07
• @Fmbalbuena : with pen and peper ;) . From the allure of the sequence I tried to understand when a higher value could appear, and then it was just try and guesses. Having so many $4$'s but no $5$ in the last number helped me understand that this was a two-steps process. Dec 30, 2021 at 13:11
• how many time do you found? Dec 30, 2021 at 13:12
• If you are asking "how much time ?", I think it was between 10 and 15 minutes. I was lucky to be on the right track from the beginning. Dec 30, 2021 at 13:14
• a 10 per 5 centimeters piece of paper (edited in the answer...) ! Dec 30, 2021 at 13:23