Silver Rectangles and the Ways of Kings

This article discusses the mathematical concept of silver rectangles, derived analogously to golden rectangles, where attaching two squares to the longer side yields a similar rectangle, leading to the silver ratio σ = 1 + √2. It then connects this ratio to central Delannoy numbers, which count the number of paths a king can take from one corner to the opposite corner of an n×n chessboard without backtracking. The article explains how the generating function for Delannoy numbers has a radius of convergence related to the silver ratio, providing an asymptotic estimate for large n. This is a technical mathematics article focused on number sequences and geometric ratios, relevant to computer science and algorithm analysis.