Visibility Degree Products in Three-Row Hexagonal Lattice Configurations

We consider lattice point configurations with rows of 2n+1, 4n+1 and 2n+1 points (3-5-3, 5-9-5, 7-13-7, ...). Two points are mutually visible if no other point of the configuration lies on the segment joining them; the degree of a point is the number of points it sees. Using the mirror symmetry of the configuration (left wing x right wing x central column) and an explicit formula for every degree, we prove a closed formula for the product of all degrees: T(n) = (4n+3)^2 (4n+4)^(4n-1) [(5n+2)^2 (5n+3)^(n-1) (5n+4)^n]^2. As a by-product, the number of mutually visible pairs is 18n^2 + 22n + 2. The formulas were checked by computer for n = 1..25. The text was prepared with the assistance of an AI model (Claude, Anthropic).