# Math sequence. What number comes next?

I have created a pattern that is fairly interesting. It is discoverable from a geometric shape. The sequence begins: 1,2,4,9,1,18,1,2.... It is a repeating pattern. It could be an infinite repeating pattern if you continue the iterations. With an infinitely high number. (This specifically is not) It's highest term is 72. It has 59 terms. What is the sequence?

• May I get a hint? Dec 9, 2020 at 21:18
• Can you reconcile "With an infinitely high number" with "It's highest term is 72"? I'm not sure I understand what the criteria of the sequence in question. Dec 10, 2020 at 20:22
• Can you also reconcile "It could be an infinite repeating pattern if you continue the iterations" with "It has 59 terms"? I'm not following the semantics here. Dec 10, 2020 at 20:23
• The sequence is based on a certain number of iterations of a function. The function can continue to iterate and the sequence would continue to grow in length and magnitude. Specifically this is based on the tenth iteration of a fractal. Dec 10, 2020 at 22:30
• Is it related to a well known fractal? Otherwise, I can pretty much artificially fit a lot of different recursive functions that satisfy your requirements. (Recursive sequences starting with 1,2,4,9,1,18,1,2... that reach at most 72 within first 59 terms.) Dec 10, 2020 at 23:17

The sequence is

1, 2, 4, 9, 1, 18, 1, 2, 36, 1, 2, 4, 72, 1, 1, 2, 4, 9, 1, 1, 2, 4, 1, 2, 1
and then
1, 2, 4, 9, 1, 18, 1, 2, 36, 1, 2, 4, 1, 2, 1, 1, 2, 4, 9, 1, 18, 1, 2, 1, 1, 2, 4, 9, 1, 1, 2, 4, 1, 2, 1

Explanation:

Here is a Heighway dragon in its tenth iteration:

Imagine the dragon is made of a wall, and we're walking along it like we're solving a maze using right-hand rule (The picture shows the beginning of our route):

And the sequence is the sizes of the rooms we encountered along the way. The "and then" in the answer corresponds to the point we reached the other side of the dragon, and started returning along the other size of the wall.

The first part of the sequence (before the "and then"), with the color of the rooms indicating their sizes:

The second part of the sequence:

Note that if the dragon is of higher iteration, the route will become longer, therefore the sequence will become longer (and contains many repeating patterns). As the iteration continues, smaller rooms will merge into larger rooms, making the largest number in the sequence higher.
Other note: I (manually) counted 60 terms in the sequence. In a long sequence like this, miscounting is very possible...