thumb|Centered hexagonal numbers appearing in the [[Catan board game:<br />19 land tiles,<br />37 total tiles]]

In mathematics and combinatorics, a centered hexagonal number, or centered hexagon number, 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:

{|style="min-width: 325px;"|

! 1 !! !! 7 !! !! 19 !! !! 37

|- style="text-align:center; color:#d20; vertical-align:middle;"

| +1 || || +6 || || +12 || || +18

|- style="vertical-align:middle; text-align:center; line-height:1.1em;"

|16px|*

|

|16px|* 16px|*<br>16px|* 16px|* 16px|*<br>16px|* 16px|*

|

|16px|* 16px|* 16px|*<br>16px|* 16px|* 16px|* 16px|*<br>16px|* 16px|* 16px|* 16px|* 16px|*<br>16px|* 16px|* 16px|* 16px|*<br>16px|* 16px|* 16px|*

|

|16px|* 16px|* 16px|* 16px|*<br>16px|* 16px|* 16px|* 16px|* 16px|*<br>16px|* 16px|* 16px|* 16px|* 16px|* 16px|*<br>16px|* 16px|* 16px|* 16px|* 16px|* 16px|* 16px|*<br>16px|* 16px|* 16px|* 16px|* 16px|* 16px|*<br>16px|* 16px|* 16px|* 16px|* 16px|*<br>16px|* 16px|* 16px|* 16px|*

|}

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 :

Formula

thumb|right|Dissection of hexagonal number into six triangles with a remainder of one. The triangles can be re-assembled pairwise to give three [[parallelograms of dots each.]]

The th centered hexagonal number is given by the formula Some examples:

{| class="wikitable defaultcenter col1left"

! Telescope !! Number of<br />segments !! Number<br />missing !! Total !! n-th centered<br />hexagonal number

|-

| Giant Magellan Telescope || &numsp;7 || 0 || &numsp;7 || 2

|-

| James Webb Space Telescope || 18 || 1 || 19 || 3

|-

| Gran Telescopio Canarias || 36 || 1 || 37 || 4

|-

| Guido Horn d'Arturo's prototype || 61 || 0 || 61 || 5

|-

| Southern African Large Telescope || 91 || 0 || 91 || 6

|}

Rocket pods

thumb|right|The M261 rocket pod has a capacity of 19 [[Hydra 70 rockets, arranged hexagonally.]]

Many rocket pods in service currently use centered hexagonal numbers of rocket tubes.

{| class="wikitable defaultcenter col1left"

|-

! Rocket Pod !! Capacity !! n-th centered hexagonal number

|-

| M260 || &numsp;7 || 2

|-

| M261 || 19 || 3

|}

References

See also

  • Hexagonal number
  • Magic hexagon
  • Star number