Editing Centered hexagonal number (section)

== Formula ==
[[Image:Centered hexagonal = 1 + 6triangular.svg|thumb|right|Dissection of hexagonal number into six triangles with a remainder of one.  The triangles can be re-assembled pairwise to give three [[parallelogram]]s of {{math|''n''(''n''−1)}} dots each.]]

The {{mvar|n}}th centered hexagonal number is given by the formula<ref name=Deza/>

:<math>H(n) = n^3 - (n-1)^3 = 3n(n-1)+1 = 3n^2 - 3n +1. \,</math>

Expressing the formula as

:<math>H(n) = 1+6\left(\frac{n(n-1)}{2}\right)</math>

shows that the centered hexagonal number for {{mvar|n}} is 1 more than 6 times the {{math|(''n'' − 1)}}th [[triangular number]].

In the opposite direction, the ''index'' {{mvar|n}} corresponding to the centered hexagonal number <math>H = H(n)</math> can be calculated using the formula

:<math>n=\frac{3+\sqrt{12H-3}}{6}.</math>

This can be used as a test for whether a number {{mvar|H}} is centered hexagonal: it will be if and only if the above expression is an integer.