On Sums of Squares Involving Integer Sequence: \(\sum_{r=1}^n w_r^2+\frac{n}{3} d^2=3\left(\frac{n d^2}{3}+\sum_{r=1}^{\frac{n}{3}} w_{3 r-1}^2\right)\)

Let wr be a given integer sequence in arithmetic progression with a common difference d. The study of diophantine equations, which are polynomial equations seeking integer solutions, has been a very interesting journey in the field of number theory. Historically, these equations have attracted the attention of many mathematicians due to their intrinsic challenges and their significance in understanding the properties of integers. In this current study, we examine a diophantine equation relating the sum of squared integers from specific sequences to a variable d: In particular, the diophantine equation \(\sum_{r=1}^n w_r^2+\frac{n}{3} d^2=3\left(\frac{n d^2}{3}+\sum_{r=1}^{\frac{n}{3}} w_{3 r-1}^2\right)\) is introduced and partially characterized. The objective is to determine the conditions under which integer solutions for (wr,d) exist within this diophantine equation.The methodology of solving this problem entails, decomposing polynomials, factorizing polynomials, and exploring the solution set of the given equation.