BBP-Type Arctangent Identities for Odd Powers of the Golden Ratio via Fibonacci–Lucas Relations — E8 Intelligence Research

FINDING: BBP-type binary expansions for arctangents of odd powers of the golden ratio, derived via Fibonacci/Lucas identities. | MATH: Let φ = (1+√5)/2 ≈ 1.6180339887. The paper (arXiv:1603.06307) derives arctan(1/φ^(2k+1)) identities using Fibonacci numbers F_n and Lucas numbers L_n. Key relations: φ^n = F_n φ + Fₙ₋₁; arctan(1/φ) = π/4 - arctan(1/φ^3) (from φ^2 = φ+1). BBP-type formula: ∑ₖ₌₀^∞ [1/(φ^(2k+1) · 2^(k+1))] · (some rational coefficient) yields binary digits of arctan(1/φ^(2k+1)). Specifically, the paper shows arctan(1/φ^(2m+1)) = ∑ⱼ₌₀^∞ (-1)^j / ( (2j+1) φ^( (2j+1)(2m+1) ) ), and then converts to base-2 BBP form using φ^(2m+1) = F₂ₘ₊₁ φ + F₂ₘ. | CONNECTION: φ is the root of x² = x+1, giving ratios 1/φ = 0.6180339887, 1/φ² = 0.3819660113 (≈0.382), φ/2 = 0.809016994, φ² = 2.6180339887. The odd powers φ^(2k+1) appear in pentagonal symmetry (cos(π/5) = φ/2, cos(2π/5) = (φ-1)/2 = 0.309016994). The BBP-type structure links φ to base-2 (binary) expansions — a direct Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com