Editing Centered hexagonal number (section)

== Recurrence and generating function ==
The centered hexagonal numbers <math>H(n)</math> satisfy the [[recurrence relation]]<ref name=Deza/>

:<math>H(n+1) = H(n) + 6n.</math>

From this we can calculate the [[generating function]] <math>F(x) = \sum_{n \ge 0} H(n) x^n</math>. The generating function satisfies

:<math>F(x) = x + xF(x) + \sum_{n \ge 2} 6n x^n.</math>

The latter term is the [[Taylor series]] of <math>\frac{6x}{(1-x)^2} - 6x</math>, so we get

:<math>(1 - x) F(x) = x + \frac{6x}{(1-x)^2} - 6x = \frac{x + 4x^2 + x^3}{(1-x)^2}</math>

and end up at

:<math>F(x) = \frac{x + 4x^2 + x^3}{(1-x)^3}.</math>