Ordering of a curious set

Having written some python to pump these out (see code here: https://repl.it/CnHE/5), I get the same sequence mentioned in the comments: \begin{matrix} \\ n & n^{th} \text{ term} & n^{th} \text{ term} \\ \hline 1 & 1 & (1+0)^2 \\ 4 & 25 & (4+1)^2 \\ 9 & 121 & (9+2)^2 \\ 16 & 361 & (16+3)^2 \\ 25 & 841 & (25+4)^2 \\ 36 & 1681 & (36+5)^2 \\ 49 & 3025 & (49+6)^2 \\ 64 & 5041 & (64+7)^2 \\ 81 & 7921 & (81+8)^2 \\ 100 & 11881 & (100+9)^2 \\ \end{matrix}

We can observe that the $n^{th}$ term is $$(n+\sqrt{n}-1)^2$$ Of course this is just an observation, not a proof...