- Centered hexagonal number
A

**centered hexagonal number**, or**hex number**, is a centeredfigurate number that represents ahexagon with a dot in the center and all other dots surrounding the center dot in ahexagonal lattice .The "n"th centered hexagonal number is given by the formula

:$1+3n(n-1).,$

Expressing the formula as

:$1+6left(\{1over\; 2\}n(n-1)\; ight)$

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

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

In base 10 one can notice that the hexagonal numbers' ones' digits follow the pattern 1-7-9-7-1.

Centered hexagonal numbers have practical applications in materials logistics management, for example, in packing round items into larger round containers, such as

Vienna sausage s into round cans.The sum of the first "n" centered hexagonal numbers happens to be "n"

^{3}. 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. In particular, prime centered hexagonal numbers arecuban prime s.The difference between (2"n")

^{2}and the "n"th centered hexagonal number is a number of the form "n"^{2}+ 3"n" − 1, while the difference between (2"n" − 1)^{2}and the "n"th centered hexagonal number is apronic number .**ee also***

Hexagonal number

*Flower of Life (generated by circles arranged in a centered hexagonal pattern)

*Star number

*Wikimedia Foundation.
2010.*

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