Centered hexagonal number

A centered hexagonal number, or hex 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 successive hexagonal layers. The centered hexagonal number for n is given by the formula

$1+3n(n+1)$ .

Expressing the formula as;

$1+6({1 \over 2}n(n+1))$

shows that the centered hexagonal number for n is the triangular number for n multiplied by 6, then add 1.

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 numbers or squares, it is necessary to solve Diophantine equations. 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 sausages into round cans.

The sum of the first n centered hexagonal numbers happens to be 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. 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 n² + 3n - 1, while the difference between (2n - 1)² and the nth centered hexagonal number is a pronic number.