hexagonal number

Meanings

1. The number of dots in a figure made up of partially overlapping dotted regular hexagons (6k dots arranged into a regular hexagon) of perimeter 6k for k = 1 ... n which grow out of a common vertex (and two shared or partially shared sides); a.k.a. “cornered hexagonal number”; H_n+1=6Δₙ-n²+1 where Δₙ denotes the nᵗʰ triangular number.
2. The number of dots in a figure made up of regular dotted hexagons (of perimeter 6k for k = 1 ... n) arranged concentrically around an extra dot at their common center; a.k.a. “centered hexagonal number”; H_n+1=1+6Δₙ.