Triangular Numbers as A2 Lattice Coordination Sequence — E8 Intelligence Research

FINDING: Triangular numbers T_n = n(n+1)/2 form the coordination sequence of the A2 root lattice (hexagonal close packing), linking discrete counting to crystallographic symmetry. | MATH: T_n = n(n+1)/2; generating function 1/(1-x)^3; A2 coordination numbers = 6, 12, 18, 24, ... = 6n (for n≥1); triangular root: n = (√(8T+1)−1)/2; "very triangular" numbers (T_n with n triangular) have density ~0 and arbitrarily long gaps (arXiv:2105.10354v2). | CONNECTION: T_n is the number of lattice points in a triangular array — exactly the A2 root system's hexagonal packing. The coordination number of A2 is 6 (first shell), matching hexagon vertices; successive shells give 6n, a linear sequence. The ratio T_n/Tₙ₊₁ → 1, but the inverse difference 1/T_n − 1/Tₙ₊₁ = 2/[n(n+1)(n+2)] — no golden ratio appears directly. However, the triangular root formula involves √(8T+1), and 8 = 2³, linking to base-2 doubling; in base-60, T_60 = 1830, and 1830/60 = 30.5 — no clean sexagesimal harmony. The A2 latti Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com