A strange sequence

This is an example of the Laurent phenomenon, a somewhat recently discovered pattern in combinatorics.

[malkimus]

from OEIS: "Any 3 successive terms of the sequence satisfy the Markov equation x^2 + y^2 + z^2 = 3 xyz. Therefore from the 3rd term on this is a subsequence of the Markov numbers, A002559. Also, we conjecture that the limit of log(log(a_n))/n is log(golden ratio)"

[jeffreyh.]

Fun fact: a similar problem (but with integer polynomials instead of just integers) appeared on 2017 Putnam A2. Interesting sequence and YT short!

[Jop_pop]

Okay I've tried for hours to see why, using various geometric concepts but I can't do it. Why is it??

Edit: it follows if you prove that if three successive terms satisfy x^2+y^2+z^2=3xyz, then the next set of three (shifted by 1 index) also does. So I've proven it but the why is still quite mystery to me

[pls_rember]

Its already strange if I dont even have a clue whatever the hell that is

[Jop_pop]

Something something induction something something