Hexagonal number

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Hexagonal number

A hexagonal number is a figurate number

Figurate number

A figurate number is a number represented as a regular and discrete geometry pattern such as a polygonal number or a polyhedral number....
, The nth hexagonal number will be the number of points in a hexagon

Hexagon

In geometry, a hexagon is a polygon with six edges and six Vertex . A regular hexagon has Schl?fli symbol ....
with n regularly spaced points on a side, as shown in .

The formula for the nth hexagonal number

The first few hexagonal numbers are:

1

1 (number)

1 is a number, number names, and the name of the glyph representing that number.It represents a single entity, the unit of counting or measurement....
, 6

6 (number)

6 is the natural number following 5 and preceding 7 .The SI prefix for 10006 is exa , and for its reciprocal atto ....
, 15

15 (number)

15 is the natural number following 14 and preceding 16 . In English, it is the smallest natural number with seven letters in its spelled name....
, 28

28 (number)

28 is the natural number following 27 and preceding 29 ....
, 45

45 (number)

45 is the natural number following 44 and followed by 46 ....
, 66

66 (number)

66 is the natural number following 65 and preceding 67 ....
, 91

91 (number)

91 is the natural number following 90 and preceding 92 ....
, 120

120 (number)

120 is the natural number following 119 and preceding 121 . 120 was known as "the great hundred", especially prior to the year 1700, from the Teutonic Hundert which equalled 120....
, 153

153 (number)

One hundred fifty-three is the natural number following 152 and preceding 154 .As a triangular number, it is the sum of the first 17 integers, and also the sum of the first five positive factorials....
, 190

190 (number)

190 is the natural number following 189 and preceding 191 .One hundred [and] ninety is a sphenic number and a Harshad number. It is a triangular number, a hexagonal number and a centered nonagonal number....
, 231, 276, 325, 378, 435, 496

496 (number)

Four hundred [and] ninety-six is the natural number following four hundred [and] ninety-five and preceding four hundred [and] ninety-seven....
, 561, 630, 703, 780, 861, 946

Every hexagonal number is a triangular number

Triangular number

A triangular number is the number of dots in an equilateral triangle evenly filled with dots. For example, three dots can be arranged in a triangle; thus three is a triangle number....
, but not every triangular number is a hexagonal number.

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Encyclopedia

A hexagonal number is a figurate number

Figurate number

Hexagon

1 (number)

1 is a number, number names, and the name of the glyph representing that number.It represents a single entity, the unit of counting or measurement....
, 6

6 (number)

6 is the natural number following 5 and preceding 7 .The SI prefix for 10006 is exa , and for its reciprocal atto ....
, 15

15 (number)

15 is the natural number following 14 and preceding 16 . In English, it is the smallest natural number with seven letters in its spelled name....
, 28

28 (number)

28 is the natural number following 27 and preceding 29 ....
, 45

45 (number)

45 is the natural number following 44 and followed by 46 ....
, 66

66 (number)

66 is the natural number following 65 and preceding 67 ....
, 91

91 (number)

91 is the natural number following 90 and preceding 92 ....
, 120

120 (number)

120 is the natural number following 119 and preceding 121 . 120 was known as "the great hundred", especially prior to the year 1700, from the Teutonic Hundert which equalled 120....
, 153

153 (number)

190 (number)

496 (number)

Triangular number

Digital root

The digital root of a number is the number obtained by digit sum, then adding the digits of that number, and then continuing until a single-digit number is reached....
in base 10 of a hexagonal number can only be 1, 3, 6, or 9.

The largest number that cannot be written as a sum of at most four hexagonal numbers is 130

130 (number)

130 is the natural number following 120 and preceding 131 ....
. Adrien-Marie Legendre

Adrien-Marie Legendre

Adrien-Marie Legendre was a France mathematician. He made important contributions to statistics, number theory, abstract algebra and mathematical analysis....
proved in 1830 that any integer

Integer

The integers are natural numbers including 0 and their negative and non-negative numberss . They are numbers that can be written without a fractional or decimal component, and fall within the set ....
greater than 1791 can be expressed in this way.

Hexagonal numbers can be rearranged into rectangular numbers n long and 2n−1 tall (or vice versa).

Hexagonal numbers should not be confused with centered hexagonal number

Centered hexagonal number

A centered hexagonal number, or hex number, is a centered polygonal number figurate number that represents a hexagon with a dot in the center and all other dots surrounding the center dot in a hexagonal lattice....
s, which model the standard packaging of Vienna sausage

Vienna sausage

A Vienna sausage is a Hot dog. The word wiener means Vienna in German language.In many European countries any pre-cooked and sometimes smoked wieners bought fresh from supermarkets, delicatessens and butcher are called Vienna sausage....
s. To avoid ambiguity, hexagonal numbers are sometimes called "cornered hexagonal numbers".

Test for hexagonal numbers

One can efficiently test whether a positive integer x is an hexagonal number by computing

If n is an integer, then x is the nth hexagonal number. If n is not an integer, then x is not hexagonal.

External links

Mathworld
MathWorld

MathWorld is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by Wolfram Research Inc. and was partially funded by the National Science Foundation's National Science Digital Library grant to the University of Illinois at Urbana-Champaign....
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