I have been looking at the sequence defined by $a_1 = 1$ and $a_{n+1} = 2a_n + \sqrt{3a_n^2 - 2}$ for any n where n is a natural number. I am trying to prove that all of the terms in the sequence are positive integers. I can prove that they're positive, but I am having trouble proving that they're integers. I tried squaring both sides to try and get rid of the root, but that didn't work. I'm not sure how to proceed.
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Have you tried proving it by induction? – John Wayland Bales Jan 27 '22 at 21:30
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1It's best to guess a formula for $a_n$ based on the first few terms, then do as @JohnWaylandBales said. Have you worked out any terms? – J.G. Jan 27 '22 at 21:31
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As JG suggests ... Calculate the firsy few values & see if you recognise the sequence https://oeis.org/A001835 – Donald Splutterwit Jan 27 '22 at 21:36