No Direct Sources Found on Apéry Recurrence or 17+12√2 Constant — E8 Intelligence Research

FINDING: Search returned only generic pedagogical material on recurrence relations and a tangential arXiv paper on graded rings of modular forms — no source directly addresses the Apéry recurrence, its characteristic polynomial, or the 17+12√2 constant. | MATH: The Apéry numbers a(n) = Σₖ C(n,k)²C(n+k,k)² satisfy the three-term recurrence (n+1)³a(n+1) = (34n³+51n²+27n+5)a(n) − n³a(n−1). The characteristic polynomial of the associated second-order operator has discriminant structure yielding the fundamental unit ε = (1+√2)² = 3+2√2, and ε² = 17+12√2. The generating function is a modular form of weight 4 on Γ₀(2) (Beukers–Zagier). | CONNECTION: 17+12√2 = (3+2√2)² = ε² where ε is the fundamental unit of ℤ[√2]. The recurrence's characteristic roots are algebraic units in the real quadratic field ℚ(√2); the ratio of dominant to subdominant root asymptotically governs a(n) ~ C·(17+12√2)ⁿ·n^(−3/2). The exponent −3/2 and the weight-4 modular form link to the Picard–Fuchs equation of a K3 surfa

Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com