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a_(n+1) = a_n - 2 if new, else a_n + 3

Source: All-Russian MO 2000

December 30, 2012
inductionalgebra

Problem Statement

The sequence a1=1a_1 = 1, a2,a3,⋯a_2, a_3, \cdots is defined as follows: if an−2a_n - 2 is a natural number not already occurring on the board, then an+1=an−2a_{n+1} = a_n-2; otherwise, an+1=an+3a_{n+1} = a_n + 3. Prove that every nonzero perfect square occurs in the sequence as the previous term increased by 33.
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