On Certain Results On The Diophantine Equation: \(\sum_{{r}={1}}^{n}w^2_r+\frac{n}{3}d^2=3(\frac{nd^2}{3}+\sum^{\frac{n}{3}}_{r=1}w^2_{3r-1})\)

Consider a sequence wr in arithmetic progression with a common difference d. he exploration of Diophantine equations, which are polynomial equations seeking integer solutions, has been a fascinating endeavor in number theory. These equations have historically intrigued mathematicians due to their inherent complexities and their importance in understanding the properties of integers. In this study, we investigate a Diophantine equation that relates the sum of squares of integers from specific sequences to a variable d. Specifically, we extend existing results on the Diophantine equation: \(\sum_{{r}={1}}^{n}w^2_r+\frac{n}{3}d^2=3(\frac{nd^2}{3}+\sum^{\frac{n}{3}}_{r=1}w^2_{3r-1})\). We aim to determine the conditions under which integer solutions for wr and d exist within this equation. Our methodology involves decomposing and factoring polynomials and exploring the solution set of the given equation.