I have decided to call this sequence $\Theta_n$ for the triangular-harmonic numbers because they clearly have properties of both triangular and harmonic numbers. The simplest closed form of the sequence is  
 \begin{equation}\Theta_i = \frac{T_{n}}{2} + \frac{n}{T_{n}^{-1}} + \frac{1}{2} \mid T_{n}^{-1} = \left \lfloor{\sqrt{2n}+\frac{1}{2}\right \rfloor \end{equation} I am currently researching its many applications.