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Find sum of all positive integers nnn such that there exist a sequence of positive integers a1,a2,...,ana_{1},a_{2},...,a_{n}a1,a2,...,an satisfying ak+1=ak2+1ak−1+1−1a_{k+1}=\frac{a_{k}^{2}+1}{a_{k-1}+1}-1ak+1=ak−1+1ak2+1−1 for every kkk with 2≤k≤n−1.2≤k≤n-1.2≤k≤n−1.
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