The ternary quadratic homogeneous equation representing homogeneous cone given by 2 2 2 X + Y = 40Z is analyzed for its nonzero distinct integer points on it . Seven different patterns of integer points satisfying the cone under consideration are obtained. A few interesting relation between the solutions and special number patterns namely Polygonal number , Pyramidal number , Octahedral number and Nasty number are presented. Also knowing an integer solution satisfying the given cone , two triples of integers generated from the given solution are exhibited.
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