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Periodic sequence Related to Golden Ratio

Source: Please Help Me !!!

July 4, 2015
number theory unsolvednumber theory with sequencesInteger sequenceperiodic functionirrational numberceiling functionnumber theory

Problem Statement

Let (an)n≥1(a_{n})_{n \ge 1} be a sequence of integers satisfying the inequality 0≤an−1+1−52an+an+1<1 0\le a_{n-1}+\frac{1-\sqrt{5}}{2}a_{n}+a_{n+1} <1 for all n≥2n \ge 2. Prove that the sequence (an)(a_{n}) is periodic.
Any Hints or Sols for this hard problem?? :help:
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