Here is the completed integer sequence.

1     11     21     1211     111221     312211     13112221     1113213211

 

The sequence is obtained by describing the previous term.

The first term is the single digit 1.

The second term describes the first term by indicating there is one 1 (11).

The third term describes the second term by indicating there are two 1s (21).

The fourth term describes the third term by indicating there is one 2 (12) and one 1 (11).

The fifth term describes the fourth term by indicating there is one 1 (11), one 2 (12), and 2 1s (21).

The sixth term describes the fifth term by indicating there are 3 1s (31), two 2s (22), and one 1 (11).

The seventh term describes the sixth term by indicating there is one 3 (13), one 1 (11),  two 2s (22), and two 1s (21).

The eight term describes the seventh term by indicating there is one 1 (11), one 3 (13), two 1s (21), three 2s (32), and one 1 (11).   

      

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