A new year indeed

Define a function \(\displaystyle f(n) = n + \left \lfloor \sqrt{n} \right \rceil\). Find the sum of all the values of \(k(\leq 2017)\) such that the composite function \(f^{k} (n) = \underbrace{f\circ f \circ f \circ \cdots \circ f}_{k \text{ times}} (n) \) is a square of integer.

Notation: \(\lfloor \cdot \rceil\) denotes the nearest integer function (round).


This problem is inspired from a Putnam problem.
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