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

Abstract

Abstract By rewriting the relation 1 + 2 = 3 {1+2=3} as 1 2 + 2 2 = 3 2 {\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 right triangles with 1 2 + 2 2 = 3 2 {\sqrt{1}^{2}+\sqrt{2}^{2}=\sqrt{3}^{2}} as initial one converging to the segment [ 0 , 1 ] {[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.