Centered hexagonal number

Centered hexagonal numbers appearing in the Catan board game:
19 land tiles,
37 total tiles

In mathematics and combinatorics, a centered hexagonal number, or hex number,[1][2] is a centered figurate number that represents a hexagon with a dot in the center and all other dots surrounding the center dot in a hexagonal lattice. The following figures illustrate this arrangement for the first four centered hexagonal numbers:

1 7 19 37
+1 +6 +12 +18
* * *
* * *
* * 		* * *
* * * *
* * * * *
* * * *
* * * 		* * * *
* * * * *
* * * * * *
* * * * * * *
* * * * * *
* * * * *
* * * *

Centered hexagonal numbers should not be confused with cornered hexagonal numbers, which are figurate numbers in which the associated hexagons share a vertex.

The sequence of hexagonal numbers starts out as follows (sequence A003215 in the OEIS):

1, 7, 19, 37, 61, 91, 127, 169, 217, 271, 331, 397, 469, 547, 631, 721, 817, 919.
  1. ^ Hindin, H. J. (1983). "Stars, hexes, triangular numbers and Pythagorean triples". J. Rec. Math. 16: 191–193.
  2. ^ Deza, Elena; Deza, M. (2012). Figurate Numbers. World Scientific. pp. 47–55. ISBN 978-981-4355-48-3.