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
$$\Theta_n = \frac{T_n^{-1}}{2} + \frac{n}{T_n^{-1}} + \frac{1}{2}$$ with $$T_n^{-1}=\lfloor\sqrt{2n}+\frac{1}{2}\rfloor$$ being the inverse triangular number function. I have been researching its many fascinating properties.