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None of the terms in the inductive sequence a_i are 0

Source: Baltic Way 1996 Q7

March 19, 2011
number theory proposednumber theory

Problem Statement

A sequence of integers a1,a2,…a_1,a_2,\ldots is such that a1=1,a2=2a_1=1,a_2=2 and for n≥1n\ge 1, an+2={5an+1−3an,if an⋅an+1 is even,an+1−an,if an⋅an+1 is odd,a_{n+2}=\left\{\begin{array}{cl}5a_{n+1}-3a_{n}, &\text{if}\ a_n\cdot a_{n+1}\ \text{is even},\\ a_{n+1}-a_{n}, &\text{if}\ a_n\cdot a_{n+1}\ \text{is odd},\end{array}\right. Prove that an≠0a_n\not= 0 for all nn.
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