A trivial observation: the following are all different ways of writing the same sequence:

\[\begin{gather} \{n^2 + n + 1 - [\sqrt{n+1}] : n \ge 0\} \\ \{(n+1)^2 - n - [\sqrt{n+1}] : n \ge 0\} \\ \{n^2 + 1 - (n + [\sqrt{n}]) : n \ge 1\} \\ \{n^2 - n + 1 - [\sqrt{n}] : n \ge 1\} \\ \{(n-1)^2 + n - [\sqrt{n}] : n \ge 1\} \\ \end{gather}\]