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Mar 27, 2019 at 14:47 comment added T. Amdeberhan @NoamD.Elkies: Thanks for this approach too.
Mar 26, 2019 at 20:00 comment added Noam D. Elkies Simpler yet: since $F_n = F_{n-1} + F_{n+2}$ it's enough to use the two-variable identity $(a+b)^3 - a^3 - b^3 = 3ab(a+b)$ which is a quick consequence of the binomial expansion of $(a+b)^3$.
Mar 26, 2019 at 19:52 history answered Cherng-tiao Perng CC BY-SA 4.0