1,2,31,2,3, some inductive real sequences and a beautiful algebraic pattern

Abstract By rewriting the relation 1 + 2 = 3 {1+2=3} as {\sqrt{1}^{2}+\sqrt{2}^{2}=\sqrt{3}^{2}} , a right triangle is looked at. Some geometrical observations in connection with plane parqueting lead to an inductive sequence of triangles initial one converging segment [ 0 , stretchy="false">] {[0,1]} real line. The their hypotenuses forms numbers which initiates some beautiful algebraic patterns. They are determined through recurrence relations proper for being evaluated computer algebra.