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So here's the sequence, that I composed on my own.

Think trees and 🟡 : 🟡

552175, 228496, 198479, 583317, 140554, 114419, 117046, ?, ?

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  • $\begingroup$ @Pdt oh I see your point, I didn't read that part very carefully before. Anyway $\varphi$ is also equally famous as $\pi$ in mathematics (geometry, especially) as I believe :) $\endgroup$
    – ACB
    Commented May 3 at 11:00
  • $\begingroup$ @ACB I of course have heard of it but I didn’t know its value. Maybe I was incorrect in saying its value isn’t on the same level as that of Pi or e. $\endgroup$
    – PDT
    Commented May 3 at 11:05

1 Answer 1

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Wow I got very lucky finding this:

Following the ‘tree’ hint I assumed that roots were somehow involved.

With a little bit of trial and error I saw $\sqrt[16]{552175}$ was 2.28496… with this promising lead I continued digging and I found $\sqrt[18]{228496}$ is in fact 1.98479…! Seeing a pattern emerge I tried several numbers for $\sqrt[?]{198479}$ and I found 3 led to 58.3317….

Then

Looking at the numbers 16,18 and 3 and after searching around I found that 1.61803 were the starting digits of the GOLDEN RATIO (1.618033988749....)!

I saw that the indexes of the roots per term were based on every successive digit pair of the golden ratio bar the 4th to 5th term (which I do believe is a mistake from the OP). The sequence I believe should then continue like so: $\sqrt[39]{583317}$, $\sqrt[88]{140554}$, $\sqrt[74]{114383}$

So how do we work out the next two terms?

We need to have $\sqrt[98]{117045}$ which is 1.12646… cashing out as 112646 and $\sqrt[94]{112646}$ cashing out as 1.13172… or 113172! Whew!

Some feedback if I may:

The puzzle would have been impossible to solve without the hint. What would have been great was if the rooting hint was somehow woven into the puzzle in the first place preferably via some flavor text. Also the knowledge tag would be great here or rather why didn’t you choose something more familiar like the digits of Pi or e?? This puzzle is already hard enough as it is.

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