Modular L-Values Link Zagier's Sporadic Sequences to Apéry Numbers — E8 Intelligence Research

FINDING: The search results are a mix of generic recurrence-solving pedagogy and one substantive research paper linking interpolated Apéry numbers to critical L-values of modular forms. The core mathematical insight is that Zagier's interpolation of Apéry numbers (for ζ(3)) is not an isolated combinatorial curiosity but a critical L-value of a weight-4 modular form, and this extends to all six of Zagier's sporadic sequences. MATH: - Apéry numbers: \( a_n = ∑ₖ₌₀^n {n}{k}^2 {n+k}{k}^2 \), satisfying recurrence \( (n+1)^3 aₙ₊₁ = (2n+1)(17n^2+17n+5) a_n - n^3 aₙ₋₁ \). - Characteristic polynomial of the recurrence (in the limit of large n, after scaling): roots are \( λ = 17 + 12√2 ≈ 33.97 \) and its reciprocal \( 17 - 12√2 ≈ 0.0294 \). Note \( 17 + 12√2 = (1+√2)^4 \), and \( (1+√2) \) is the silver ratio. - Modular form connection: The interpolated Apéry numbers \( A_t(n) \) (Zagier's t-deformation) are shown to Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com