To install click the Add extension button. That's it.

The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time.

4,5
Kelly Slayton
Congratulations on this excellent venture… what a great idea!
Alexander Grigorievskiy
I use WIKI 2 every day and almost forgot how the original Wikipedia looks like.
Live Statistics
English Articles
Improved in 24 Hours
Added in 24 Hours
Languages
Recent
Show all languages
What we do. Every page goes through several hundred of perfecting techniques; in live mode. Quite the same Wikipedia. Just better.
.
Leo
Newton
Brights
Milds

# Centered hexagonal number

A centered hexagonal number, or hex number,[1] 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. Centered hexagonal numbers have practical applications in materials logistics management.

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

## Description

Dissection of hexagonal number into six triangles with a remainder of one. The triangles can be re-assembled pairwise to give three parallelograms of n(n−1) dots each.

A centered hexagonal 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.

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

The nth centered hexagonal number is given by the formula

${\displaystyle n^{3}-(n-1)^{3}=3n(n-1)+1.\,}$

Expressing the formula as

${\displaystyle 1+6\left({\frac {n(n-1)}{2}}\right)}$

shows that the centered hexagonal number for n is 1 more than 6 times the (n − 1)th triangular number.

The first few centered hexagonal numbers are (sequence A003215 in the OEIS):

1, 7, 19, 37, 61, 91, 127, 169, 217, 271, 331, 397, 469, 547, 631, 721, 817, 919.

## Properties

In base 10 one can notice that the hexagonal numbers' rightmost (least significant) digits follow the pattern 1–7–9–7–1.

The sum of the first n centered hexagonal numbers is n3. That is, centered hexagonal pyramidal numbers and cubes are the same numbers, but they represent different shapes. Viewed from the opposite perspective, centered hexagonal numbers are differences of two consecutive cubes, so that the centered hexagonal numbers are the gnomon of the cubes. (This can be seen geometrically from the diagram.) In particular, prime centered hexagonal numbers are cuban primes.

The difference between (2n)2 and the nth centered hexagonal number is a number of the form 3n2 + 3n − 1, while the difference between (2n − 1)2 and the nth centered hexagonal number is a pronic number.

## Applications

Centered hexagonal numbers have practical applications in materials logistics management, for example, in packing round items into larger round containers, such as Vienna sausages into round cans, or combining individual wire strands into a cable.

## Finding the root

The root n of a centered hexagonal number x can be calculated using the following formula:

${\displaystyle n={\frac {3+{\sqrt {12x-3}}}{6}}}$

## References

1. ^ Hindin, H. J. (1983). "Stars, hexes, triangular numbers and Pythagorean triples". J. Rec. Math. 16: 191–193.