Centered Pentagonal Numbers II: Geometric Sequence Visualization

This example demonstrates how centered pentagonal numbers are formed by repeatedly expanding concentric, center-based pentagonal layers, creating a visual sequence that can also be compared with frequency-style distributions. The key mathematics includes centered polygonal numbers (the centered version of pentagonal numbers), sequence generation (recursive/explicit relationships), and discrete data visualization such as histogram-like rectangular bars and frequency distribution. Construction highlights from the GeoGebra command structure: Sequence is used to generate the sequence terms, while Zip bundles elements into drawable data sets; Element selects specific indexed terms/objects for the current layer; Text adds numerical annotations. Mod can control periodic behaviors or grouping of indices during the animation. Functions/parameters such as f, cx, cy set the mapping from values to geometry and define the center coordinates; N, LN, pVp, L_1, L_2, and n determine the layer count, ranges, and scaling. The size variable scales graphical primitives. Numerous point/object arrays (A1–A21, B1–B6) provide vertices and auxiliary points for building each centered pentagon layer. Suitable for upper-secondary and early undergraduate lessons on number patterns, geometric sequences, and interactive exploration in GeoGebra.