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Lets have two sequences.
S1 12,45,171,648,2457,9315,35316 and
S2 21,81,306,1161,4401,16686,63261
The ratio of the last two terms 63261/35316 is almost equal to $\sqrt[3]{3(\sqrt{21}-1)-5}$. If we want a closer approximation we can obtain more terms of the above sequences. The question is h...