An Irrational Fibonacci-Like Sequence in Base 7 Converging to phi_7 = (1 + sqrt(11/7)) / 2 — A Structured Alternative to the Golden Ratio

This work introduces a Fibonacci-like sequence defined in base 7, converging to the irrational constant phi_7 = (1 + sqrt(11/7)) / 2 — a structured analogue distinct from the classical golden ratio. Each term follows the recurrence H_n = H_{n-1} + (1/7) * H_{n-2}, with initial values H_0 = 7 and H_1 = 8. The sequence produces exact fractions of the form k / 7^n, aligned with base-7 numeration and prime-length jumps. These terms reveal a harmonic and predictable structure based on rational coefficients. The limit phi_7 satisfies the minimal equation x^2 - x - 1/7 = 0 and generalizes across all bases using the root of x^2 - x - 1/b. Unlike the classical Fibonacci sequence, this one maintains arithmetic alignment with its base and generates fractal-like rational patterns. This construction offers new perspectives in recursive arithmetic, base-specific growth systems, and geometric representations grounded in exact rational structure.