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Prove the identity of sequence - ISL 1976

Source:

September 20, 2010
algebraIMO Shortlistrecurrence relationpower of 2Sequencefloor functionimo 1976

Problem Statement

A sequence (un)(u_{n}) is defined by u_{0}=2   u_{1}=\frac{5}{2}, u_{n+1}=u_{n}(u_{n-1}^{2}-2)-u_{1}   \textnormal{for } n=1,\ldots Prove that for any positive integer nn we have [un]=2(2n−(−1)n)3 [u_{n}]=2^{\frac{(2^{n}-(-1)^{n})}{3}} (where [x] denotes the smallest integer ≤\leq x)..
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