Appendix Four
The principle behind Gijswijt’s sequence is to look for repeating blocks of numbers in the previous terms of the sequence. The ‘block’ has to be at the end of the sequence of previous terms, and the number of times it is repeated provides the next term.
Mathematically, the sequence is described as follows. Start with a 1, and then each subsequent term is the value k, when the previous terms are multiplied in order and written xy k for the largest possible value of k.
The sequence is 1, 1, 2, 1, 1, 2, 2, 2, 3, 1, 1, 2, 1, 1, 2, 2, 2, 3, 1, 1…
I think it is easiest to understand what is at work here by considering the first time a 3 appears, which is in position 9. The previous terms multiplied in order are 1 × 1 × 2 × 1 × 1 × 2 × 2 × 2. What Gijswijt requires us to do is to transform this sum into a term xy k for the largest value of k. In this case, we get (1 × 1 × 2 × 1 × 1) × 23. So, the following term is a 3. We are looking for the largest repeating block of numbers at the end of the sequence of previous terms, although in this case the block is a single number, 2, repeated three times.
But often the block will have several digits. Consider position 16. The previous terms multiplied tther are 1 × 1 × 2 × 1 × 1 × 2 × 2 × 2 × 3 × 1 × 1 × 2 × 1 × 1 × 2. This can be written (1 × 1 × 2 × 1 × 1 × 2 × 2 × 3) × (1 × 1 × 2)2. So, the 16th term is a 2.
Going back to the beginning now, the second term is a 1 since the previous term 1 is not multiplied by anything. The third term is a 2 since the previous terms multiplied in order are 1 × 1 = 12, and the fourth term is 1 since the previous terms result in (1 × 1 × 2) × 1, where the final 1 is not multiplied by itself.