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3, 8, 24, 72, 213, 668, 2064, 6312, ...

What's the next number?

This sequence mod 3 is 0, 2, 0, 0, 0, 2, 0, 0;

mod 2 is 1, 0, 0, 0, 1, 0, 0, 0;

mod 5 is 3, 3, 4, 2, 3, 3, 4, 2.

And there seems to be no patterns mod anything else. I could now conclude that the next number $a_9$ should have the form of $30n+3$.

It can be also seen that $a_{n+1}/a_n \approx 3$. In fact, the sequence could be written like $$3^1,\ 3^2-1,\ 3^3-3,\ 3^4-9,\ 3^5-30,\ 3^6-61,\ 3^7-123,\ 3^8-249,\ \cdots$$ The remainders are $$0, -1, -3, -9, -30, -61, -123, -249, \cdots$$ Mod 30 we have $$0, -1, -3, -9, 0, -1, -3, -9, \cdots$$ Therefore $a_9 - 3^9 \equiv 0 \mod 30.$ But this may not be helpful as it offers no information other than $a_9 = 30n+3$. And I am afraid that I've gone too far off the track.

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  • $\begingroup$ Is this a puzzle you found that you don't have an answer for? The would bbe helpful to know, as then we don't have to expect a hint. What is the source of the puzzle? That's helpful to guide us in how to tackle the puzzle $\endgroup$ – P1storius Sep 6 '19 at 14:34
  • $\begingroup$ It's from a math problem set. And what I've written here is just my attempt to solve it. I will remove it if it's misleading or violating the code of conduct. $\endgroup$ – Ze Chen Sep 6 '19 at 14:42
  • $\begingroup$ The remainder sequence goes $\times 2 +1$, $\times 2 +1$, $\times 2 +3$, $\times 3 +3$, repeat. $\endgroup$ – Arnaud Mortier Sep 6 '19 at 20:28