Right Triangles with Lucas Number Legs and Their Expressions via Metallic Means

Building upon the author’s earlier geometric framework in which pairs of right triangles — one with legs from a generalized Fibonacci (integer) sequence Gn and the other with legs from the corresponding Lucas companion sequence Ln — encode powers of a Metallic Mean δm through their angle sums and catheti ratios, this paper introduces a new and complementary discovery. We show that right triangles whose both legs are taken from the classical Lucas number sequence yield hypotenuses H = √(L_m² + L_n²) that admit elegant closed-form expressions in terms of higher-order Metallic Means δk. Three distinct algebraic forms are identified and classified by a small integer surplus r = H² − ⌊H⌋²: a subtraction form (r = 1), a fractional-part form (r = 4), and a square-shift form (r = 8). Exact Diophantine identities are established, and a remarkable meta-metallic nested structure is uncovered for the pair (76, 199). These findings deepen the geometric bridge between Lucas sequences, right-triangle geometry, and the full family of Metallic Means.