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

Abstract

By rewriting the relation 1+2=3 as √12+√22=√32 , a right triangle is looked at. Some geometrical observations in connection with plane parqueting lead to an inductive sequence of right triangles with √12+√22=√32 as initial one converging to the segment [0,1] of the real line. The sequence of their hypotenuses forms a sequence of real numbers which initiates some beautiful algebraic patterns. They are determined through some recurrence relations which are proper for being evaluated with computer algebra.