The Metallic Right-Triangles

Abstract

The Kepler triangle and its kin are systematically described by the unique formalism (Sugimoto, 2020). The present study affords the super-set of the formalism covering the above-mentioned triangles and other new triangles based on the metallic means and the generalised Fibonacci sequences adjoined to those means. The super-set is given by (the short leg, the long leg, the hypotenuse) = (G_{n - 2}^1/2(m), m^1/2Φ^n/2(m), G_n^1/2(m)Φ(m)), where G_n(m) and Φ(m) designate the nth generalised Fibonacci sequence and the metal mean of the metal number 'm', respectively. The initial members of the golden, silver and bronze right-triangles are presented.