In the sequence formed by concatenating the digits 1, 2, 3, ..., i, i+1, ..., n what is the first appearance of k repetitions of the same digit?

For example, when k=3 the answer is:
11, 12: 111 (3 1s in a row).

Show with the same format for k = 2, 4, 5.
How far can you go?
Is there a pattern?

Note: k means "exactly k", so for k=2, the answer is not 11 because the next digit is another 1.