Hexagonal number

Hexagonal number

A hexagonal number is a figurate number, The "n"th hexagonal number will be the number of points in a hexagon with "n" regularly spaced points on a side, as shown in [http://mathworld.wolfram.com/HexagonalNumber.html] .

The formula for the "n"th hexagonal numbe

h_n= n(2n-1),!

The first few hexagonal numbers OEIS|id=A000384 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 not every triangular number 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.

Any integer greater than 1791 can be expressed as a sum of at most four hexagonal numbers, a fact proven by Adrien-Marie Legendre in 1830.

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

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".

Test for hexagonal numbers

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

:n = frac{sqrt{8x+1}+1}{4}.

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

External links

*Mathworld entry on [http://mathworld.wolfram.com/HexagonalNumber.html Hexagonal Number]


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