Centered k-gonal numbers: C(k,n) = 1 + k·n(n-1)/2. A central point plus k·triangular layers. Centered hex (k=6): 1,7,19,37,61,91... = 3n²−3n+1 — counts hex grid cells. The 'hex numbers' appear in chemistry (benzene rings) and game boards.
How to Use
Choose polygon sides k, enter n:
- C(k,n): The centered number
- Formula: 1 + k·T(n-1)
- Layers: Visual layer counts
Hex Numbers
Centered hexagonal (k=6): 1,7,19,37,61,91,127... = 3n²−3n+1. These are cube differences: n³−(n-1)³. They appear in hex grids (board games like Settlers of Catan, Go variants) and honeycomb structures.
Chemistry
Centered hexagonal numbers count atoms in polycyclic aromatic hydrocarbons: benzene (7), coronene (19), circumcoronene (37). The hex grid structure reflects carbon bonding geometry.
Step-by-Step Instructions
- 1Choose polygon sides.
- 2Enter layer number.
- 3Compute centered number.
- 4View layers.
- 5Compare shapes.