The sextic non-homogeneous equation with four unknowns represented by the Diophantine equation is(4k-1)(x^2+y^2 )-(4k-2)xy=4(4k-1)z^6 analyzed for its patterns of non-zero distinct integral solutions are illustrated. Various interesting relations between the solutions and special numbers, namely polygonal numbers, Pyramidal numbers, Jacobsthal numbers, Jacobsthal-Lucas number are exhibited.