Inmathematics andcombinatorics, acentered hexagonal number, orcenteredhexagon number,^[1][2] is acenteredfigurate number that represents ahexagon with a dot in the center and all other dots surrounding the center dot in ahexagonal lattice. The following figures illustrate this arrangement for the first four centered hexagonal numbers:

1		7		19		37
+1		+6		+12		+18

Centered hexagonal numbers should not be confused withcornered hexagonal numbers, which are figurate numbers in which the associated hexagons share a vertex.

The sequence of hexagonal numbers starts out as follows (sequenceA003215 in theOEIS):

Thenth centered hexagonal number is given by the formula^[2]

${\displaystyle H(n)=n^3-(n-1{)}^3=3n(n-1)+1=3n^2-3n+1.}$

Expressing the formula as

${\displaystyle H(n)=1+6\left(\frac{n(n-1)}{2}\right)}$

shows that the centered hexagonal number forn is 1 more than 6 times the(n − 1)thtriangular number.

In the opposite direction, theindexn corresponding to the centered hexagonal number${\displaystyle H=H(n)}$ can be calculated using the formula

${\displaystyle n=\frac{3+\sqrt{12H-3}}{6}.}$

This can be used as a test for whether a numberH is centered hexagonal: it will be if and only if the above expression is an integer.

The centered hexagonal numbers${\displaystyle H(n)}$ satisfy therecurrence relation^[2]

${\displaystyle H(n+1)=H(n)+6n.}$

From this we can calculate thegenerating function${\displaystyle F(x)=\sum _{n\ge 0}H(n)x^n}$. The generating function satisfies

${\displaystyle F(x)=x+xF(x)+\sum _{n\ge 2}6n{x}^n.}$

The latter term is theTaylor series of${\displaystyle \frac{6x}{(1-x{)}^2}-6x}$, so we get

${\displaystyle (1-x)F(x)=x+\frac{6x}{(1-x{)}^2}-6x=\frac{x+4x^2+x^3}{(1-x{)}^2}}$

and end up at

${\displaystyle F(x)=\frac{x+4x^2+x^3}{(1-x{)}^3}.}$

Inbase 10 one can notice that the hexagonal numbers' rightmost (least significant) digits follow the pattern 1–7–9–7–1 (repeating withperiod 5).This follows from thelast digit of the triangle numbers (sequenceA008954 in theOEIS) which repeat 0-1-3-1-0 when taken modulo 5.Inbase 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 firstn centered hexagonal numbers isn³. That is, centered hexagonalpyramidal numbers andcubes 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 thegnomon of the cubes. (This can be seen geometrically from the diagram.) In particular,prime centered hexagonal numbers arecuban primes.

The difference between(2n)² and thenth centered hexagonal number is a number of the form3n² + 3n − 1, while the difference between(2n − 1)² and thenth centered hexagonal number is apronic number.

Manysegmented mirrorreflecting 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.^[3] Some examples:

• Hexagonal number
• Magic hexagon
• Star number

• Figurate numbers
• Integer sequences

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