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a_1 = 1, a_n = n - 2 if $a_{n-1} = 0 and a_n = a_{n-1} - 1 , otherwise

Source: 2016 Saudi Arabia IMO TST , level 4, III p1

July 29, 2020
Sequencealgebrarecurrence relation

Problem Statement

Define the sequence a1,a2,...a_1, a_2,... as follows: a1=1a_1 = 1, and for every n≥2n \ge 2, an=n−2a_n = n - 2 if an−1=0a_{n-1} = 0 and an=an−1−1a_n = a_{n-1} - 1, otherwise. Find the number of 1≤k≤20161 \le k \le 2016 such that there are non-negative integers r,sr, s and a positive integer nn satisfying k=r+sk = r + s and an+r=an+sa_{n+r} = a_n + s.
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