Three Derivations of Binet's Formula via Sums, Generating Functions, and a Fibonacci Tree — E8 Intelligence Research

FINDING: Multiple derivations of Fibonacci closed form (Binet's formula) via binomial sums, generating functions, and a novel Fibonacci-tree representation that computes F_n independently of prior terms. MATH: - Binet's formula: \(F_n = {φ^n - (-φ)⁻ⁿ}{√5}\), where \(φ = {1+√5}{2} ≈ 1.6180339887\). - Binomial sum identity (from generating functions / Wilf): \(F_n = {1}{√5}∑ₖ₌₀n/2 {n}{2k+1} 5^k\) — equivalent to Binet but expressed via binomial coefficients. - Fibonacci tree representation (arXiv:1302.6583): \(F_n = ∑T ∈ T_n 1\) where \(T_n\) is the set of Fibonacci trees of order \(n\); closed form via tree enumeration without recursion — essentially a combinatorial bijection to binomial sums. - Generating function: \(∑n≥0 F_n x^n = x/1-x-x^2\), whose partial fraction decomposition yields Binet. CONNECTION: - \(φ\) and its inverse \(1/ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com