in reply to Re^2: Conway's audioactive sequence oneliner
in thread Conway's audioactive sequence oneliner
It is made up only of 1's, 2's, and 3's
We can prove this by contradiction. Because the sequence starts with '1', the only way a '4' can show up is as the first number in a pair (the "run" value -- the bold values in 312211). After all, that's the only way that the '2' and '3' ever show up. So this means we must have a sub-sequence such as "1111", "2222", or "3333" in our sequence. Let's abstract these as "xxxx".There are two ways "xxxx" can be placed in the sequence, at an even offset or an odd offset. At an even offset, the first and third 'x's are counts; at an odd offset, the second and third 'x's are counts. Let's examine the the even offset first. You can't have "C1xC2x" in the sequence, because that means it should have been encoded as "(C1+C2)x". Similarly, at an odd offset, there must be a count before the first 'x' (we'll call it C1 again), which means we have "C1xxxx" in our sequence. Again, you can't have two counts in a row for the same value! The subsequence would have to be "(C1+x)xx".
So this means there will never be four like values in a row, thus '4' will never be in this sequence. (You can prove that '2' and '3' WILL be in the sequence.)