Does this sequence even exist?

Find sum of all positive integers nn such that there exist a sequence of positive integers a1,a2,...,ana_{1},a_{2},...,a_{n} satisfying ak+1=ak2+1ak1+11a_{k+1}=\frac{a_{k}^{2}+1}{a_{k-1}+1}-1 for every kk with 2kn1.2≤k≤n-1.

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