On the non-homogeneous quintic equation with five unknowns

The non-homogeneous quintic equation with five unknowns represented by x³ + y³ = z³ + w³ +6T⁵ is considered. Three different patterns of infinitely many non-zero integral solutions of the above the quintic equation are presented. Various interesting realations between the solutions and special number patterns,namely, Polygonal numbers, Star numbers, Centered Polygonal, Jacobsthal lucas numbers and Jacobsthal numbers are exhibited.