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.
Highlights
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
Summary
By introducing positive numbers rk by r2k = m2k − 1, k ∈ N, a sequence of circles |z − mk| = rk in the complex plane is given which is part of some parqueting of the complex plane [4]. The parqueting-reflection principle is always initiating iteratively given sequences of complex numbers; see [2] for another sample. A. Abdymanapov et al, 1, 2, 3, inductive real sequences and a beautiful algebraic pattern. Abdymanapov et al, 1, 2, 3, inductive real sequences and a beautiful algebraic pattern | 247 is seen. Combining this with in the definitions for αk+1 and βk+1 shows mk+2. Lemma 1.2 suggests a new definition of the sequence with just one initial value m1 = √3 and mk+1.
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