The Centered hexagonal number reference article from the English Wikipedia on 24-Apr-2004 (provided by Fixed Reference: snapshots of Wikipedia from wikipedia.org)

Centered hexagonal number

A centered hexagonal number, or hex number is a figurate number that represents a hexagon with a dot in the center and all other dots surrounding the center dot equidistantly. The centered hexagonal number for n is given by the formula

.

Expressing the formula as;

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

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

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

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.

See also regular hexagonal number.

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