Centered Hexagonal Number

1+3n(n+1).\,

Expressing the formula as

:

1+6\left({1\over 2}n(n+1)
ight)

shows that the centered hexagonal number for ''n'' is 1 more than 6 times the ''n''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

All centered hexagonal numbers are odd, and in base 10 one can notice the one's digits follows the pattern 1-7-9-7-1.

To find centered hexagonal numbers besides 1 that are also Triangular Number s or squares, it is necessary to solve Diophantine Equation s. By solving the Diophantine equation

:

{1 \over 2} m(m + 1) = 3n^2 + 3n + 1

we learn that 91, 8911 and 873181 are numbers that are both centered hexagonal numbers and triangular numbers (they grow very large after that), while solving the Diophantine equation

:

m^2 = 3n^2 + 3n + 1\,

we learn that 169 and 32761 are centered hexagonal numbers that are also squares.

Centered hexagonal numbers have practical applications in materials logistics management, for example, in packaging 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 are Cuban 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 a Pronic Number .

See also regular Hexagonal Number , and the Flower Of Life pattern (generated by circles arranged in a centered hexagonal pattern).

Information About

Centered Hexagonal Number