Can I find the Pseudoinverse (Moore-Penrose inverse) just by knowing the one-sided inverses of a matrix?

Question

Consider a matrix such as $B = \begin{bmatrix} 1 & 0 & 2 \\ 0 & 1 & 1 \end{bmatrix}$. I know how to compute the right inverses (or in the case of $m\geq n$ the left inverses) and have done so; I've obtained the result $B^{-1}_{R} = \begin{bmatrix} 1-2c_{1} & -2c_{2} \\ -c_{1} & 1-c_{2} \\ c_{1} & c_{2} \end{bmatrix}$. However, I now want to calculate the (unique) Moore-Penrose Pseudoinverse, preferably using this right-sided inverse. Clearly, it would have to be one of the right-sided inverses. Using matlab I've found that the Moore-Penrose Pseudo inverse equals $B_{R}^{-1}$ for $c_{1} = \frac{1}{3}, c_{2} = \frac{1}{6}$. Is there a way I could easily get those correct values of $c_{1}$ and $c_{2}$ (by that I mean without using a tool such as Matlab)?