Sylvester's Sequence: Doubly-Exponential Growth and Recurrence Essence — E8 Intelligence Research

FINDING: Sylvester's sequence (OEIS A000058) exhibits a doubly-exponential growth with a self-similar structure in its denominator differences, but the search results are mostly pedagogical videos; the core mathematical essence is the recurrence and its asymptotic constant. MATH: - Definition: \( a_0 = 2 \), \( aₙ₊₁ = a_0 a_1 ⋯ a_n + 1 = a_n(a_n - 1) + 1 \) (since \( a_n = a_0⋯ aₙ₋₁+1 \)). - Growth: \( a_n ≈ E2^n + 1/2 \) where \( E ≈ 1.264084735305301 \) (the "Sylvester constant"). - Key identity: \( 1/a_0 + 1/a_1 + ⋯ + 1/a_n = 1 - 1/a_0 a_1 ⋯ a_n \), so the infinite sum of reciprocals = 1. - Denominator differences: \( aₙ₊₁ - a_n = a_n^2 - 2a_n + 1 = (a_n - 1)^2 \). Thus the difference is a perfect square — a self-similar squaring structure. CONNECTION: - The recurrence \( aₙ₊₁ = a_n^2 - a_n + 1 \) is a quadratic map. Its fixed points satisfy \( x = x^2 - x + 1 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com