The descriptor sequence is a sequence of numbers in which the digits of each number describe the preceding number. The first number is 1. This number consists of one 1, so the second number is 11 (that is, one-one). This consists of two 1's, so the third term is 21. This consists of one 2 and one 1, so the fourth term is 1211. The first six numbers in the sequence are:
1, 11, 21, 1211, 111221, 312211.
Show that no digit greater than 3 ever occurs, and that the string 333 never occurs.
The numbers in the even positions of a number that is a descriptor cannot have the same number twice in a row. That is because if a descriptor number has abcb, then the number that it describes has a b's, c b's. Then, those 2 runs of b's are really part of the same run, so the descriptor would have db, where d>=a+c. Therefore, if you take the numbers at the even positions of a descriptor number, then no number repeats twice in a row. Then, no digit can repeat more than thrice in a row in a descriptor number. If it repeated 4 or more times, then at least 2 in a row would be at even positions. Since the most a digit can repeat in a descriptor is thrice, then the biggest possible digit in the descriptor of the descriptor is 3. Therefore, no digit >3 ever occurs.
Suppose 333 is in a number in the Descriptor Sequence. Then, the number it describes must have 3 3's, so it has 333. Then, the number that number describes has 333. Then, every number before it in the sequence has 333. That is a contradiction because it must hit 1. Therefore, 333 never occurs.
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Posted by Math Man
on 2024-06-08 16:13:05 |
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