Centered hexagonal number

# Centered hexagonal number

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Encyclopedia
A centered hexagonal number, or hex number, is a centered figurate number
Figurate number
The term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes and different dimensions...

that represents a hexagon with a dot in the center and all other dots surrounding the center dot in a hexagonal lattice
Hexagonal lattice
The hexagonal lattice or equilateral triangular lattice is one of the five 2D lattice types.Three nearby points form an equilateral triangle. In images four orientations of such a triangle are by far the most common...

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1 7 19 37
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The nth centered hexagonal number is given by the formula

Expressing the formula as

shows that the centered hexagonal number for n is 1 more than 6 times the (n−1)th triangular number
Triangular number
A triangular number or triangle number numbers the objects that can form an equilateral triangle, as in the diagram on the right. The nth triangle number is the number of dots in a triangle with n dots on a side; it is the sum of the n natural numbers from 1 to n...

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The first few centered hexagonal numbers are
1, 7, 19
19 (number)
19 is the natural number following 18 and preceding 20. It is a prime number.In English speech, the numbers 19 and 90 are often confused. When carefully enunciated, they differ in which syllable is stressed: 19 vs 90...

, 37
37 (number)
37 is the natural number following 36 and preceding 38.-In mathematics:It is a prime number, the fifth lucky prime, the first irregular prime, the third unique prime and the third cuban prime of the form...

, 61
61 (number)
61 is the natural number following 60 and preceding 62.-In mathematics:It is the 18th prime number. The previous is 59, with which it comprises a twin prime. Sixty-one is a cuban prime of the form p = / , x = y + 1....

, 91
91 (number)
91 is the natural number following 90 and preceding 92.-In mathematics:Ninety-one is the twenty-seventh distinct semiprime and the second of the form...

, 127
127 (number)
127 is the natural number following 126 and preceding 128.- In mathematics :*As a Mersenne prime, 127 is related to the perfect number 8128. 127 is also an exponent for the Mersenne prime 2127 - 1, making 127 a double Mersenne prime...

, 169
169 (number)
169 is the natural number following 168 and preceding 170.-In mathematics:* 169 is an odd number* 169 is a square number. 13 x 13 = 169, and the reverse 31 x 31 = 961...

, 217, 271, 331, 397, 469, 547, 631, 721, 817, 919.

In base 10 one can notice that the hexagonal numbers' ones' digits follow the pattern 1-7-9-7-1.

Centered hexagonal numbers have practical applications in materials logistics management, for example, in packing
Packing problem
Packing problems are a class of optimization problems in mathematics which involve attempting to pack objects together , as densely as possible. Many of these problems can be related to real life packaging, storage and transportation issues...

round items into larger round containers, such as Vienna sausage
Vienna sausage
A Vienna sausage is a kind of hot dog...

s into round cans, or combining individual wire
Wire
A wire is a single, usually cylindrical, flexible strand or rod of metal. Wires are used to bear mechanical loads and to carry electricity and telecommunications signals. Wire is commonly formed by drawing the metal through a hole in a die or draw plate. Standard sizes are determined by various...

strands into a cable
Cable
A cable is two or more wires running side by side and bonded, twisted or braided together to form a single assembly. In mechanics cables, otherwise known as wire ropes, are used for lifting, hauling and towing or conveying force through tension. In electrical engineering cables are used to carry...

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The sum of the first n centered hexagonal numbers happens to be n3. That is, centered hexagonal pyramidal numbers and cubes
Cube (arithmetic)
In arithmetic and algebra, the cube of a number n is its third power — the result of the number multiplying by itself three times:...

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
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

centered hexagonal numbers are cuban primes.

The difference between (2n)2 and the nth centered hexagonal number is a number of the form 3n2 + 3n − 1, while the difference between (2n − 1)2 and the nth centered hexagonal number is a pronic number
Pronic number
A pronic number, oblong number, rectangular number or heteromecic number, is a number which is the product of two consecutive integers, that is, n . The n-th pronic number is twice the n-th triangular number and n more than the n-th square number...

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• Hexagonal number
Hexagonal number
A hexagonal number is a figurate number. The nth hexagonal number will be the number of points in a hexagon with n regularly spaced points on a side.The formula for the nth hexagonal number...

• Flower of Life
Flower of Life
The Flower of Life is the modern name given to a geometrical figure composed of multiple evenly-spaced, overlapping circles. They are arranged to form a flower-like pattern with a sixfold symmetry, similar to a hexagon...

(generated by circles arranged in a centered hexagonal pattern)
• Magic hexagon
Magic hexagon
A magic hexagon of order n is an arrangement of numbers in a centered hexagonal pattern with n cells on each edge, in such a way that the numbers in each row, in all three directions, sum to the same magic constant. A normal magic hexagon contains the consecutive integers from 1 to...

• Star number
Star number
A star number is a centered figurate number that represents a centered hexagram, such as the one that Chinese checkers is played on.The nth star number is given by the formula 6n + 1...