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# Hexagonal number

A hexagonal number is a figurate number. The nth hexagonal number hn is the number of distinct dots in a pattern of dots consisting of the outlines of regular hexagons with sides up to n dots, when the hexagons are overlaid so that they share one vertex.

The formula for the nth hexagonal number

${\displaystyle h_{n}=2n^{2}-n=n(2n-1)={{2n}\times {(2n-1)} \over 2}.\,\!}$

The first few hexagonal numbers (sequence A000384 in the OEIS) are:

1, 6, 15, 28, 45, 66, 91, 120, 153, 190, 231, 276, 325, 378, 435, 496, 561, 630, 703, 780, 861, 946...

Every hexagonal number is a triangular number, but only every other triangular number (the 1st, 3rd, 5th, 7th, etc.) is a hexagonal number. Like a triangular number, the digital root in base 10 of a hexagonal number can only be 1, 3, 6, or 9. The digital root pattern, repeating every nine terms, is "1 6 6 1 9 3 1 3 9".

Every even perfect number is hexagonal, given by the formula

${\displaystyle M_{p}2^{p-1}=M_{p}(M_{p}+1)/2=h_{(M_{p}+1)/2}=h_{2^{p-1}}}$
where Mp is a Mersenne prime. No odd perfect numbers are known, hence all known perfect numbers are hexagonal.
For example, the 2nd hexagonal number is 2×3 = 6; the 4th is 4×7 = 28; the 16th is 16×31 = 496; and the 64th is 64×127 = 8128.

The largest number that cannot be written as a sum of at most four hexagonal numbers is 130. Adrien-Marie Legendre proved in 1830 that any integer greater than 1791 can be expressed in this way.

Hexagonal numbers can be rearranged into rectangular numbers of size n by (2n−1).

Hexagonal numbers should not be confused with centered hexagonal numbers, which model the standard packaging of Vienna sausages. To avoid ambiguity, hexagonal numbers are sometimes called "cornered hexagonal numbers".

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## Test for hexagonal numbers

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

${\displaystyle n={\frac {{\sqrt {8x+1}}+1}{4}}.}$

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

## Other properties

### Expression using sigma notation

The nth number of the hexagonal sequence can also be expressed by using Sigma notation as

${\displaystyle h_{n}=\sum _{i=0}^{n-1}{(4i+1)}}$

where the empty sum is taken to be 0.

### Sum of the inverse of hexagonal numbers

Sum of the inverse of hexagonal numbers is 2ln(2). ln is Natural logarithm.

{\displaystyle {\begin{aligned}\sum _{k=1}^{\infty }{\frac {1}{k(2k-1)}}&=\lim _{n\to \infty }2\sum _{k=1}^{n}\left({\frac {1}{2k-1}}-{\frac {1}{2k}}\right)\\&=\lim _{n\to \infty }2\sum _{k=1}^{n}\left({\frac {1}{2k-1}}+{\frac {1}{2k}}-{\frac {1}{k}}\right)\\&=2\lim _{n\to \infty }\left(\sum _{k=1}^{2n}{\frac {1}{k}}-\sum _{k=1}^{n}{\frac {1}{k}}\right)\\&=2\lim _{n\to \infty }\sum _{k=1}^{n}{\frac {1}{n+k}}\\&=2\lim _{n\to \infty }{\frac {1}{n}}\sum _{k=1}^{n}{\frac {1}{1+{\frac {k}{n}}}}\\&=2\int _{0}^{1}{\frac {1}{1+x}}dx\\&=2[\ln(1+x)]_{0}^{1}\\&=2\ln {2}\\&\approx {1.386294}\cdots \\\end{aligned}}}