Here's an execution for 2, 3. When a != 1 we use a as the extra numbers instead of b:
 1 | 2 2(1) ...
 2 | 2 2(0) 2(1) ...
 3 | 3 2(1) 2(0) 2(1) ...
 4 | 3 2(0) 2(0) 2(1) ...
 5 | 2 3(2) 2(1) 2(0) 2(0) ...
 6 | 2 3(1) 2(1) 2(0) 2(1) ...
 7 | 2 3(0) 2(1) 2(0) 2(1) ...
 8 | 3 3(2) 2(0) 2(0) 2(1) ...
 9 | 3 3(1) 2(0) 2(0) 2(1) ...
10 | 3 3(0) 2(0) 2(0) 2(1) ...
11 | 2 2(1) 3(2) 2(1) 2(0) 2(1) ...
12 | 2 2(0) 3(2) 2(1) 2(0) 2(1) ...
13 | 3 2(1) 3(1) 2(1) 2(0) 2(1) ...
14 | 3 2(0) 3(1) 2(1) 2(0) 2(1) ...
15 | 2 2(1) 3(0) 2(1) 2(0) 2(1) ...
16 | 2 2(0) 3(0) 2(1) 2(0) 2(1) ...
17 | 3 3(2) 3(2) 2(0) 2(0) 2(1) ...
Furthermore, note that if a = 1, then the a, b sequence is a subset of the b, a sequence e.g.:
1, 2 = [1, 2, 2, 1, 1, 2, 1, ...]
2, 1 = [   2, 2, 1, 1, 2, 1, ...]
therefore we can always make a not be 1 by switching the pair and then using the generalized algorithm with a != 1.

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  1. Kolakoski sequence
  2. Integer sequence
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