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A set of three elements is called arithmetic if one of its elements is the arithmetic mean of the other two. Likewise, a set of three elements is called harmonic if one of its elements is the harmonic mean of the other two.

How many three-element subsets of the set of integers  $\left\{z\in\mathbb{Z}\mid -2011<z<2011\right\}$  are arithmetic and harmonic?

(Remark: The arithmetic mean  $A(a,b)$  and the harmonic mean  $H(a,b)$  are defined as
\[A(a,b)=\frac{a+b}{2}\quad\mbox{and}\quad H(a,b)=\frac{2ab}{a+b}=\frac{2}{\frac{1}{a}+\frac{1}{b}}\mbox{,}\]
respectively, where  $H(a,b)$  is not defined for some  $a$ ,  $b$ .)

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