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Interesting Recursive Integer Sequence

Source: India Postal Coaching 2015

December 2, 2015
SequenceinductionRecurrencenumber theory

Problem Statement

The sequence <an><a_n> is defined as follows, a1=a2=1a_1=a_2=1, a3=2a_3=2, an+3=an+2an+1+n!an,a_{n+3}=\frac{a_{n+2}a_{n+1}+n!}{a_n}, n≥1n \ge 1. Prove that all the terms in the sequence are integers.
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