On a Conjecture on the Representation of Positive Integers as the Sum of Three Terms of the Sequence $\lfloor \frac{n^2}{a}\rfloor$

We prove some cases of a conjecture by Farhi on the representation of every positive integer as the sum of three terms of the sequence $\lfloor \frac{n^2}{a}\rfloor$. This is done by generalizing a method used by Farhi in his original paper.