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

Abstract: 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 $ \left\lfloor\frac{n^2}{a}\right\rfloor$. This is done by generalizing a method used by Farhi in his original paper.