All rep'n'digits have infinite subsequence, except the rep'n'digits 3 (mod 9) and 6 (mod 9).
For 'n' is 1, we have the Kaprekar numbers (A145875), the repdigit numbers.
If length is 1 (mod 9), repdigit 1 is part of the sequence,
1111111111*1111111111 = 1234567900987654321 => 123456790 + 987654321 = 1111111111.
If length is 2 (mod 9), repdigit 5 is part of the sequence,
55555555555*55555555555 = 3086419753024691358025 => 30864197530 + 24691358025 = 5555555555.
If length is 4 (mod 9), repdigit 7 is part of the sequence, 7777 * 7777 = 60481729 => 6048 + 1729 = 7777.
If length is 5 (mod 9), repdigit 2 is part of the sequence.
If length is 7 (mod 9), repdigit 4 is part of the sequence.
If length is 8 (mod 9), repdigit 8 is part of the sequence.
Repdigit 9 is part of this sequence in every length.
For 'n' is 2, we have numbers where two digits are repeated, like 52525252.
The rep2digits which are divisible by 9 have the following infinite subsequences:
If length is 2 (mod 22), rep2digit 54 is a part of this sequence, 545454545454545454545454 * 454545454545454545454545 = 247933884297520661157024297520661157024793388430 => 247933884297520661157024 + 297520661157024793388430 = 545454545454545454545454
If length is 4 (mod 22), rep2digit 27 is a part of this sequence.
If length is 6 (mod 22), rep2digit 18 is a part of this sequence.
If length is 8 (mod 22), rep2digit 63 is a part of this sequence.
If length is 10 (mod 22), rep2digit 90 is a part of this sequence.
If length is 14 (mod 22), rep2digit 36 is a part of this sequence.
If length is 16 (mod 22), rep2digit 81 is a part of this sequence.
If length is 18 (mod 22), rep2digit 72 is a part of this sequence.
If length is 20 (mod 22), rep2digit 45 is a part of this sequence.
Other rep2digits also have infinite subsequences with length l (mod 198).
Example: rep2digit 52 has length 8: 52525252 * 25252525 = 1326395239261300 => 13263952 + 39261300 = 52525252, the next length is 206.
