Centered hexagonal number

Centered hexagonal number
In mathematics and combinatorics, 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 a hexagonal lattice. The following figures illustrate this arrangement for the first four centered hexagonal numbers:
{|style="min-width: 325px;"|
! 1 !! !! 7 !! !! 19 !! !! 37
|- style="text-align:center; color:red; vertical-align:middle;"
| +1 || || +6 || || +12 || || +18
|- style="vertical-align:middle; text-align:center; line-height:1.1em;"
|
|
|
|
|
|
|
|}
Centered hexagonal numbers should not be confused with cornered hexagonal numbers, which are figurate numbers in which the associated hexagons share a vertex.
The sequence of hexagonal numbers starts out as follows :
1, 7, 19, 37, 61, 91, 127, 169, 217, 271, 331, 397, 469, 547, 631, 721, 817, 919.
Formula
The th centered hexagonal number is given by the formula
Expressing the formula as
shows that the centered hexagonal number for is 1 more than 6 times the th triangular number.
In the opposite direction, the index corresponding to the centered hexagonal number can be calculated using the formula
This can be used as a test for whether a number is centered hexagonal: it will be if and only if the above expression is an integer.
Recurrence and generating function
The centered hexagonal numbers satisfy the recurrence relation
From this we can calculate the generating function . The generating function satisfies
The latter term is the Taylor series of , so we get
and end up at
Properties
In base 10 one can notice that the hexagonal numbers' rightmost (least significant) digits follow the pattern 1–7–9–7–1 (repeating with period 5).
This follows from the last digit of the triangle numbers which repeat 0-1-3-1-0 when taken modulo 5.
In base 6 the rightmost digit is always 1: 16, 116, 316, 1016, 1416, 2316, 3316, 4416...
This follows from the fact that every centered hexagonal number modulo 6 (=106) equals 1.
The sum of the first centered hexagonal numbers is . 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 and the th centered hexagonal number is a number of the form , while the difference between and the th centered hexagonal number is a pronic number.
Applications
Centered hexagonal numbers have practical applications in packing problems. They arise when packing round items into larger round containers, such as Vienna sausages into round cans, or combining individual wire strands into a cable.
Many segmented mirror reflecting telescopes have primary mirrors comprising a centered hexagonal number of segments (neglecting the central segment removed to allow passage of light) to simplify the control system. Some examples:
References
See also
Hexagonal number
Magic hexagon
Star number
Figurate numbers
Integer sequences